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Full text of "A Treatise on arithmetic in theory and practice [microform] : for the use of schools"

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Serica of National 0cl)ool Book 



TREATISE ON ARITHMETIC, 



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THEORY AND PEAC'JICE. 



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FOR 



^%^ UlSit (9(5 ^I^DjlUiVa^. 



Aathorh-d hy the Council of Public Instruction, 
for Upper Canada. 



TORONTO: 
PUnLISIlI^^D BY ROBERT MclMIAIL, 
65, KTiNG Strket East. • 
1860. * 



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I' K E F A U JK 




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In Iho prosont edition a vast number of eicrcisM I,.,. 

be,n «ldod, that no ,„,e, however trifling nlTL ,oft 

w.thout ,0 many ill„,t™tiona as should f rvet ml u 

uftc,ently fi.^.har t. the pupi,. And when it was feared 

m-glit not at oneo suggest itself; some question calculated 

ocrs and " the pnnoiples of notation and numeration - 
for the teacher may re.t assured, that the facility, I^d e^^ 
the success, wUh which subsequent pa>-ts of his tostrucZ 
wirbe .onveycd to the mind of the learner, dependsinl' 
great degree, upon an adequate acquaintance wHh ti 
Hence, to proceed without scouring a perfect and pra^S 
knowledge of this part of the suyect, is to retard'TaS 
than to accelerate improvement. 

. ?» '■"P''-.*'™"' "« ^--y oommencemcnt, must be m.>da 

due d. Of the great utility of teehnica'. iangua..e faecut 
tately „ndersto«l) it is almost superfluous to say anvZv 
hero : we cannot, however fni.l,o..,. ^i ■ ' ""y'hmg 

cillin,, t,,.. • ""we\oi, iMbear, upon this oocas on, re- 
call ng to ren,cmbra«ee wh.at is so admirably and so effeo. 

J.ven n the common mchanical arts, something of a 
t.chn,cal language is 6,md needful f„r th<L. who .arMcaru? 



'* PREFACE , 

ing or exercising them. It would be a very groat la 
convenicnpo, even to a common carpenter, not to have a 
precise, well understood name for each of the several opera- 
tiona he performs, such as chiselling, sawing, planing, ^^c, 
md for the several tools [or instruments] he works with. 
And If we had not such words as addition, subtraction, 
multiplication, division, &c., employed in an exactly defined 
sense, and also fixed rules for conducting tho.e and other 
arithmetical processes, it would be a tedious and uncertain 
work to go through even such simple calculations as a child 
very soon learns to perform with perfect ease. And after 
all there would be a fresh difficulty in making other per- 
sons understand clearly the correctness of the calculations 
made. 

"You are to observe, however, that technical Ian.rua.^o 
and rules, if you would make them really useful, must be not 
only (bshndly urulerstood, but also learned and yemembered as 
famiharly as the alphabet, and employed cmshmtlu, and 
With Born^uhns eimtnm; otherwise, technical langua-e will 
p.-ove an encumbrance instead of an advantage, just as a 
suit of clothes would b. if, instead of putting thorn on and 
wearing them, you.were to carry thorn about in your hand " 
Page 11. 

What is said of technical lan^riMge is, at least, equally true 
0. the signs and characters by which we still further tae'ilitate 
the conveyance of our ideas on such matters as form the 
subject of the present work. It is mucli more simple to put 
down a character whicu expresses a process, than to write 
the name, or description of the latter, in fuJl. Besides, in 
glancing over a mathematical investigation, the mind is 
able, with greater ease, to connect, and understand its dif- 
ferent portions when they are briefly expressed by familiar 
Bigns, than when they are indicated by words vdnch havo 
nothing particularly calculated to catch the eije, and which 
cannot even be clearly undcrstocKi without considerable 
attention. But it mu«c be bortie in mind, that, while such 
«■ trsatise as the present, will seem easy and intelligible 



▼a 



PREFACE 

enough if M,e „ign,, ^Uch it contains in almost evorr na™ 

mo or loss obscure to those who have not been habitfS 
t he „so them. They are, however, so few and so L^H 
that there is no eicnse for their not hoin^ perfcctlv onir 
stood-partienlarly by the teacher of arithme:;o ' '''■ 
hhould peculiar e.rcumstanccs render a different arrange- 

■ IT ' u ° ■"■''"'"' "' ■«"^'' ""i^^aWe, the j„dicio,« 
master will never be at lo8« how to act^there «« Z 

ntelhgcnco of tho pupil, will render it necessary to conflno 

u .nstrucfon to the more important branches. 4, i^^r 

Bhould, If possible, mako it an inviolable rule to receive 

rerorVetr """"'"ri' "^ "^ e^pWion r 
reason. The references which have been subjoined to tha 

different questions and which indicate the paJgiphs wh re 

neswrh "'' ''"^*'" be obtained, and alsfthL^r 
enees which are scattered throagh the work, will, be fould 
of considerable assistance ; for, as the most i^teUi-nt pnTu 
wi 1 oecasionally forget something he has learned he^a^ 

e.ple" ■ ""' '™ '»™"*^' *'"'™ '"e has seen it 
I)«™<,& have been treated of at the same time as integers 
be^^ause since both of them follow ,re,set, the same ifw"' 
when the rales relating to integers are fullv understood 
here IS „o&V,. „» to be learn:! on the sub/eelrpt^ ct 
larly If what has been said with reference to numeratirand 
notation is carefully borne in mind. Should it, however "n 

unti the learner shall have made some further advance. 
_ ilie most useful portions of menlai arithmetic have been 
ntrodueed into "Practice- and the other rules with whch 
they seemed more immediately connected. 

the mind of the toner and when he is found to have been 



^" PRKFACE. 

guilty of any inaccuracy, ^o Bhould be made to correct him 
»elf by repeating each part of the appropriate rule, and 
oxeniphfyin- it, until ho perceives hia error. It should be 
continually kept in view that, in a work on such a subject 
as arithmetic, any portion must seem difficult and obscura 
without a knowledge of what precedes it. 

The table of logarithms and article on the subject, also 
the table of squares and cubes, squarr. roots and cube roota 
of numbers, which have been introduced at the end of the 
work, will, It IS expected, prove very acceptable to the more 
•dvanced arithmetician. 



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CONTENTS 



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PART I. 

Multiplication Table, . . . , 

Tables of money, weights, and mcasuree, 
Definitions, • . . . 

Section I.— Notation and Numeration, 
Arabic system of numbers, 
lionian notation, . 

' • • • 

Skction II. -^Simple Addition, 
To prove Addition, 

Addition of quantities containing decimals, 
Simple Subtraxjtion, 
To prove Subtraction, 

iSubtraction of quantities containing decimals, 
Simple Multiplication, . 
To multiply when neitlier multiplicand nor mul 
tiplier exceeds 12, . 

When the multiplicand exceeds 12, 

To prove Multiplication, 

To multiply quantities, when there are cyphers 
or decimals, 

When both multiplicand and multiplier ex 

eee«^. 12, 

To prove Multiplication, 

To prove MultipHcation, by casting out the nines, 

To multiply, 60 as to have a certain number of 

decimal places in the product, 
To multiply by a composite number, 

~ by a number not composite. 

To multiply by a number consisting of nines. 
Simple Division, . . 

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CONTENTS 



when tho divisor does not exceed 12, 
Mividend 12 times the divisor, 
♦1, V .J ".'" *^'^'^°'" ^*^* ""* "oeed 12, bufc 

4:ot:s:r"*^^^ 

To divide when the dividend, divisor, or both' 

contain cyphers or decimals, 
- When the divisor exceeds 12, 
To prove Division, . * ' 

To divide by a composite number, 

^^^^^ ^L^ "'""^'''^ ^"- ""^« J^« *han one 
expressed by unity and one or more cyphers, 

lofind the greatest common measure of nimberi 
10 tnd their least common multiple 
SECTION III.-Reduction Descending, ' ' 

Reduction Ascending, . . * * * 

To prove Reduction, . ^,' 

The Compound Rules, . .' * ' 

' Compound Addition, . ,' ' ' 

Compound Subtraction, 

Compound Multiplication, when the multiplier 
does not exceed 12, . . 

• When the multiplier exceeds 12 and ig 
composite, . . «"iu is 

— - When the multiplier is the sum of compo^ 
Bite numbers, . . ^ 

-— When the multiplier is not composite, .' 

Compound Division, when the divisor la abstract, 

and does not exceed 12, . 

■ When the divisor exceeds 12 and' is com. 
pgsite. 

When the divisor exceeds 12 and is not com 
posite, 

-— When the divisor and di vid'end are both apl 
plicate, but not of the same denomination • or 
more than ono 'lenomination is found in either 
or both, 






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COMTKNTi 



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%(:tion IV.— Vulgar Fractions, 
\ Toreducoun improper fraction to a mixed n 

To reduce un integer to a fraction, 

To reduce fractions to lower terms, 

To find the value of a fraction in terms of a lower 
denoiaination, . 

To express one quantity as the fraction of another, 

To add fractions having a common denominator, 

To add fractions when their denominators are 
different and prime to each other. 

To add fractions having different denominators, 
not all prime to each other. 

To reduce a mixed number to an improper frac- 
tion. 

To add mixed numbers, . 

' • • 

To subtract fractions which have a commoh 

denominator, . 

■ • • • 

To subtract fractions which have not a commoa 

denominator, . . . . , 

To subtract mixed numbers, or fractions from 

mixed numbers, 
To multiply a fraction and whole number together 
To multiply one fraction by another, 
To multiply a fraction, or mixed number by 

mixed number, 
To divide a fraction by a whole number, 
To divide a fraction by a fraction, 
To divide a whole number by a fraction, 
To divide a mixed number by a whole number or 

fraction. 
To divide an 'nteger by a mixed number, 
To divide a fraction or mixed number by a mixed 

number, 

' • • • 

►Vheu the divisor, dividend, or both, are com. 

pound, or complex fractions, . 
Oeoimal Fractions, .... 

To reduce a vulgar fraction to a decimal or to a 

decimal fraction. 
To reduce a decimal to a lower denomination, 



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CONTENTS. 



To find at <me the decimal equivaJent to any 
number of shillings, pence, &o., ^ 

Circulating Decimals, . * * * 

To change a circulating decimal into its equi* 

vaJent vulgar fraction, ^ 

When a vulgar fraction will give a finite decimal,' 
Thenumber of decimal places in a finite decimal 
The number of digits iu the period of a circulate 
— When a circulating decimal will contain a 

hnite part, . . 

Contractions in multiplication a«d division, de 
rived from the properties of fractions, 
Section^ \.— Proportion, . 

Natiu^ of ratios, . , '' 

Nature of Proportion, 

To fil *^' ^jthmetical mean of* two quantities,' 

To find a fourth proporti. al, when the first term 

isumty, . . ^ "* 

— When .ho second or third term is imity, 
To find the geometrical mean of two quantities,- 
Fr operties of a geometrical proportion, . 
Rule of Simple Proportion, 

— When the first and second terms are not of 
.he sa:ne or contain different denominations, 

— When the tbu-d term contains more than one 
denomination, ... 

If fractions or mixed numbJrs are found in 

any of the terms, 
Piule of Compound Proportion, 
To abbreviate the process, ' . 



PART n. 
&CTI0N VI.~Practice, 

To find aliquot parts, . . * * 

To 2nd the price of one denomination, ttiat of a 
higher being given, , 



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CONTENTS. 

To find the price of more than one lower denomi- 
nation, • . . , 
To find the price of one higher denomination, .' 
To find the price of more than one higher deno- 
mination, 

Given the price of one denomination, to find that 

of any number of another, . 
When the price of any denomination is the aliquot 
part of a shilling, to find the price of any num- 
ber of that denomination, 
When the price of any denomination is the aU- 
quot part of a pound, to find the price of any 
number of that denomination, . 
When the complement of the price, but not the 
price itself, is the aliquot part or parts of a 
pound or shilling. 
When neither the price nor its complement is the 

aliquot part or parts of a pound or shilling. 
When the price of «ach article is an even number 
of shiliin./A to find the price of a number of 
ai'ticieK 

* • • • 

When the price is an odd numbor and loss 

than 20, . . 

To find the price of a quantity reproscited by a 
mixed number, 

Given the price per cwt., to find that of cwt., 
qrs., &c., 

Given the price per pound, to find that of cwt.,* 

^qrs., &c., 

Given the price per pound, to find that of a ton. 
Given the price per ounce, to find that of ounces, 
pennyweights, &c., . 

Given the price per yard, to find that of yards,' 
qrs., &«., 

To find the price of acres, roods, &o., . 
Given the price per quart, to find thet of a hoc*- 

tead. ". 

Given the price per quart, to find that of a tun, 



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CONTENTS 



Given the pri^e of one article in pence, to find 

that of any number, . 
Given wages per day, to find their amount per 

year, . . ^ ^ . 

Bills of parcels, . . ; • . 

Tare and Tret, . ' * * 

• • • . 

Section Vll.-Simple Interest. 

To find the simple interest on any sum, for a year, 
— When the rate per cent, consists of more 

than one denomination, 
To find the interest on any sum for years* 

For years, months, &o., 

To find the interest on any sum, for a^y time," 
at .5, b, &c., per cent., . 

When the rate, or number of years, or both,' 
are expressed by a mixed number. 
To find the interest for days, at five per cent., .* 
To find the interest for days, at any other ratq, . 
10 find the interest for months, afr 6 per cent , 
To find the interest of money left after one or 
more payments. 

Given the amount, rate, and time~to find the 
principal, 

Given the time, rate, and principal-to find the 
amount, 

Given the amount, principal, and rate— to find 
the time. 

Given the amount, principal, and time,' to find 
the rate. 

Compound Interest-given the principd, rate,' 

f '.d time— to find the amount and interest. 
To find the present worth of any sum. 

Given the principal, rate, and amoJnlrlto find 
the time. 

Discount, 

• • • 

To find discount, • , , 

To compute Commission. lnsiirnTift« 'p««i,^-„«» 



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CCNTENT8. 

To find what insurance must be paid that, if the 
goods lire lost, both their value and the insur 
ance paid may be recovered, .. 
Purchase of Stock, 
Equation of Payments, . 
Sjcction VIII.— -Exchange, . 
Tables of foreign money. 
To reduce bank to current money, 
To reduce current to bank money. 
To reduce foreign to British money, 
To reduce British to foreign money, 
To reduce florins, &c., to pounds, &c., Flemish, 
To reduce pounds, &o., Flemish, to florins, &c., 
Simple Arbitration of Exchanges, 
Compound Arbitration of Exchanges, 

To estimate the gain or loss per cent.. 

Profit and Loss, . 

• • • 

To find the gain or loss per cent.. 
Given the cost price and gain— to find the selling 
price, . . . . ^ 

Given the gain or loss per cent., and the selling 

price*-to find the cost price, . 
Simple Fellowship, 

Compound Fellowship, . . , ' 

Barter, 

Alligation Medial, 
Alligation Alternate, 

When a given amount of the mixture is re 

quire,d, 

When the amount of one ingredient is given. 

Sjcction IX.— Involution, . 

To raise a number to any power. 

To raise a fraction to any power. 

To raise a mixed number to any power, 

Evolution, 

To find the square root, . 

When the square contains decimals., 

To find the square root of a fraction, 



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CONTENTS. 



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To and the square root of a miiod number, 
10 IincI the cube root, 

'T7 Y^®" *^^ ^^^^^ contains decimals, ' 

lo find the cube root of a fraction. 

To find the cube root of a mixed number,* 

1 extract any root whatever, 

To find the squares and cubes, the squm-o and 

cube roots of numbers, by the table, . 
Logarithms, . 

\tle, ^^' ^'^^''^^'^ "^ ^ given numbed by th 

To find the logarithm of a fraction. 

To find the logarithm of a mixed numbed, 

log^iim, "^"'^^ ^^^;'^^'^^"^^"« *« ^ Siven 
— If the given logarithm is not in the' table, * 
lo multiply numbers by means of their loga-* 

To divide numbers by melns of their logarithms* 

l^^t^^^^^^^^^P--^^-^^ 
To evolve a quantity by means ofits logarithm. ' 
SECTION X.— On Progression, 

To find the sum of a series of terms in arUhmeti-' 

cal progression, • 

In an arithmetical series given the extremes and 
number of terms-to find the common diS 
ence, , ^ 

\^.t r """"^'^ '^ arithmet'ical m;ans,be: 
tween two given numbers, 

^'s fries,'"^ ^''''^'"^'' '"'"^ '^ '-^"y '^^^'itl^neticai 

^"cZ&'7-f'"^ ''"'^' *S™ '^' ^^*^-«*"^- and 
common difference-tofindthenumberof terms, 

~r. '\*^' '"™ "^ "^« ««^i««' "^e number of 

Tn ni' ''"" «^treme-to find the other, . 

To find the sum of a series of terms in geometri- 

oal progression, . b^ometri- 



rage 

313 
313 
315 
316 
31G 
' 317 

318 
319 

321 
323 
323 

324 
324 

325 

326 

327 
327 



329 



329 



330 



ooi 



332 
332 
333 






CONTENTS. 

Ih a geoulGtrical series, given the extremes and 
number of term&— to find the common ratio, 

To find any number of geometrical means between 
two quantities, 

To find any particular term of a geometrical series, 

In * geometrical series, given the extremes and 
common ratio— to find the number of terms, . 

In a geometrical series, given the common ratio, 
the number of terms, and one extreme— to find 
the other • . . . 

Annuities, . . 

* ■ • 

To find the amount of a certain number of pay- 
ments in arrears, and the interest due on them, 

To find the present value of an annuity, 

When it is in perpetuity, . 

To find the value of an annuity in reversion, 

Position, . 

5"ingle Position, . 

Double Position, . 

Miscellaneous exercises, 
Table of Logarithms, 

Table of squares and cubes, and of square and 
cube roots, «... 

Table of the amounts of £1, at compound interest] 

Table of the amounts of an annuity of £1, 

Table of the present values of an annuity of ^1, 

Irish converted into British acres, 
' Value of foreign money in British, . [ 



Tago 
335 

336 
336 

337 



338 
340 

340 
342 
343 
344 
345 
346 
347 
355 
361 

377 

385 
385 
386 
386 
386 



T5/ EATLSi: ON ARITHMETIC: 



IM 



THEORY AND PRACTICE. 



ARITHMETIC. 



PART I. 



TABLES. 

MULTIPLICATION TABLE. 



|i 



Twice 
1 are 2 
2—4 
3 *- 6 

4-8 

5 — 10 

6 — 12 

7 - 14 

8 — 16 

9 — 18 

10 — 20 

11 — 22 

12 — 24 



8 times 
1 are 3 
2—6 
8—9 



12 
15 
18 
21 
24 



9 — 27 
10 — 80 
n — 83 
12 — 36 



4 times 
1 are 4 
2—8 

3 -r- 12 

4 — 16 

5 — 20 
6—24 

7—28 
8—32 



9 
10 
11 
12 



86 
40 
44 
48 



5 times 

1 are 5 

2 — 10 

3 — 15 

4 — 20 

6 — 25 

6 — 30 

7 — 35 

8 — 40 

9 — 45 

10 — 60 

11 — 55 

12 — 60 



6 times 
1 are 6 
2—12 

3 — 18 

4 — 24 
6 — 30 
6 — 36 
7—42 

8 — 48 

9 — 54 

10 — 60 

11 — 66 
12 



7'^ 



7 times 

1 are 7 

2 — 14 
8—21 
4 — 28 
5—35 
6—42 
7—49 

8 — 66 

9 — 63 

10 — 70 

11 — 77 

12 — 84 



8 times 


1 


are 8 


2 


— 16 


3 


— 24 


4 


— 32 


5 


— 40 


6 


— 48 


7 


— 56 


8 


— 64 


9 


— 72 


10 


— 80 


11 


— 88 


12 


— 9S 



9 times 

1 are 9 

2—18 

3—27 

4—36 

6—45 

6—64 

7 — 63 

8—72 

9—81 

10 — 90 

11 — 99 

12 — 108 



10 times 

1 are 10 

2—20 

3—30 

4—40 

6—60 

6 - 

7 - 

8 - 
9—90 

10 — 100 

11 — 110 

12 — 120 



60 
70 
80 



11 times 

1 are 11 

2—22 

3—33 

4—44 

5 — 55 

6—66 

7—77 

8—88 

9—99 

10 — 110 

11 — 121 

12 — 132 



12 times 
1 are 12 
2—24 
3—36 
4 — 48 I 
6 — 60 I 

6 — 72 I 

7 — 84 , 

8 — 96 i 

9 — 108 ! 

10 — 120 \ 

11 — 132 

12 — li4 



It appears from tliis table, that the multiplication of tho 
same two uumbers, m whatever order taken, produces tlio 



SIGNS USED IN THIS TREATISE. 





' 


7 times i 


1 


are 7 


2 


— 14 


3 


— 21 


4 


— 28 


5 


— 35 


6 


— 42 


7 


— 49 


8 


— 66 


9 


— 63 


10 


— 70 


n 


— 77 


12 


— 84 



12 times 


1 


are 12 


2 


- 24 


3 


- 86 


4 


— 48 


6 


— 60 


e 


— 72 , 


7 


- 84 . 


8 


— 96 ; 


9 


~ 108 ! 





— 120 I 


1 


— 132 


2 


- Ui 


ion of tho 


iucca tlw 



^^ + the sign of addition; as 5+7, or 5 to bo addc4 

tra7ted''froT4"^ "*'''°"™ ' "^ ^-^' "' ^ '» ^' »»''. 
muHiptVbfg"' -"'"J"''''"'-'" i »» 8X9, or 8 to be 
^^•^ the sign of division ; as 18+6, or 18 to be divided 

ti,„ ft? vinculum, which is used to show that all 
the quanfafe s united by it are to be oonsiderld as but 

rbe^^n^r^^^^I^l-ral^e^-f^p^HeltJ 

thl|ts'lTssfh™*i.""'" "■"* ' '^ S>-«^'er than J, and 

thJ rati^of frZ/'i"- '" '■f"".™' *"' 5:6, means 
me ratio ot j to 6, and is read 5 is to 6 

: : indicates the eqiiality'of ratios ; thus, 5 : 6 • • 7 • 8 

means that there is the slm. relation bet'ween 5 'and 6 

as between 7 and 8 ; and is read 5 is to 6 as 7 is ?o 8 

y the radical sign. By itself, it is the sign of the 

2:\t rootff 4', or 4^ it "^^ ^' '' '' '' ^^' '^ 

641-31, &c. may be read thus : taE 3 f?om 8, add 7 to thJ 
difference, multiply the sum by 4, divide the nroduct hv fi 
take the square root of the quotient Td to °t ^dd 31 tLn 
multiply the sum by the cube root of 9, divide tho product 

^ — r' — '•' ••'- product will bo equal to 041-31. &c 
1 hesc «sns arc /tUly cq^laincd in tla-ir proper places. 



I 



MULTIl'LICATIOiN TAIJI.E. 



eaino result; tlius I times 0, and G times 5 jiro 30:— tho 
reason will bo oxplainod when we treat of multiplication. 
Ihero are, therefore, several repetitions, which, although 
many persons conceive tlioin unnecessary, are not, perhaps, 
quite unprofitable. Tho following is free from such an 
objection : — 



f 



Twice 2 
» 3 
4 
6 
6 
7 
8 
9 



>> 
>> 
•> 
•> 
II 
II 



are 4 

— 6 

— 8 
~ 10 
~ 12 

— 14 

— 18 

— 18 



8 times 8 
4 
6 
6 

7 
8 
9 



II 
II 
II 
II 
II 
II 



9 
12 
15 
18 
21 
24 
27 



6 times 7 are 35 
8 — 4a 
9—45 



II 



6 times 6 
7 
8 
9 



II 
II 

II 



86 
42 
48 

64 



4 times 4 
6 
6 
7 
8 




II 
•I 
II 
II 



16 
20 
24 
28 
82 
86 



5 times 6 — 26 
II 6—30 



7 times 7 
II 8 
1, 9 


— 49 

— 66 

— 63 


8 times 8 
., 9 


— 64 

— 72 


9 times 9 


81 



10 times 8 are 80 
II 9 — 90 
I, 10 —100 
„ 11 —110 



10 times 2 are 20 



— 80 

— 40 

— 60 
6 — 60 
7—70 



8 
4 
6 



11 times 2 
8 
4 
6 
6 
7 
8 
9 



II 
II 
II 
II 
II 
II 
11 



22 
83 
44 

55 
66 
77 
88 
99 



12 times 2 — 24 
„ 8 — 86 
4-48 

5 60 

6 - 72 
7—84 
8—96 
9 —108 

10 —120 

11 —182 

12 —144 

I 



II 
II 
II 
II 
II 



"Ten," or "eleven times," in the above, scarcely requiion 
to be committed to memory; since we perceive, that to 
multiply a number by 10, we have merely to add a cypher to 
tho right hand side of it :— -thus, 10 times 8 are 80; and to 
multiply it by 11 wo have only to set it down twice :-~thus. 
11 times 2 rtro 23. 



TADLE OF MONEY. 



a 



2 — 22 


8 — 83 


i — 44 


5 — 55 


3 — 66 


J — 77 


B — 88 


) — 99 


2 — 24 


J — 86 


t - 48 


) 00 


5 - 72 


' - 84 


5 — 96 


> —108 


> —120 


— 182 


1 —144 




7 requirfl.i 


3, that to 


cypher to 


0', and to 


se : — thus, 



Tho following tables aro required* for reduction, tho 
compound rules, &c., and may bo committed to memory 
us convcmonce suggests. ^ 

TABLE OF MONEY. 
A farthing ia the smallest coin generally used in this 
country, it is represented bj . . . /""*"'^' 

raako 1 halfpenny, i 
1 penny, d, 
1 shilling, » 



Kaitlilngg 
o 



4 or 

48 

900 
1,008 



halt'ponco 



24 or 

480 
504 



pence 
12 



I shillings 
240 or 20 
252 or I 21 



1 pound, 
1 guinea. 






The symbols of pounds, shillings, and ponce, are placed 

over the numbers which express them. Thus 3 14 o" 
means, tliree pounds fourteen shillings, and sixpence'.' So'ilic! 
times only the symbol for pounds is used, an'd is placed 

hofore the whole quantity ; thus, £3 „ 14 „ G. 3 9^ moans 
t ireo 8.nlhno-8 and mnepence halfpenny. 2s. 6?./. means two 
faulhngs and sixpence three farthinf^n, &c 

When learning the aI)ove and foflowing tables, the pupil 
HhMuld be ri^uired, at hr.st, to commit to memory only those 
pn-t urns which are over the thick angular linos; thusf in tho 
one just given:— 2 farthings make one halfpenny; 2 half- 
pence one penny; 12 pence one shilling; 20 shillings one 
pound; and 21 shillmgs one guinea. a ° 

Ih h I'oaUy mean the quarter, half, and three quarters 
of a penny, d. is used as a symbol, because it is the first 



S. 

2 

5 

13 



6 make one half Crown. 
one Crown. 

4 one Mark. 



I 



4 WEIflHTa. 

AVOIRDUPOISE WEiailT. 
Its name ia derived from French-— and ultimately 
from Latin words signify innj " to have weight." It u 
used in weighing heavy articles 

Drams 

10 ... . 



2oO ar 

7,168 

28,672 



448 or 
1,792 
673,4401 85,840 



ouncei 
16 



pound* 
28 



112 or 
2,240 



quarteri 

4 



SymboU 

make 1 ounce, oz. 
. 1 pound, lb. 
. 1 quarter, q. 
lhuudrc(l,cwt. 



hundred! 

lA IK ■^- 80 or I 20 .1 ton. t. 

J4 lbs., and m Bomo cases 16 lbs., make 1 stone. 
20 stones . . . 1 barrel. 



TROY WEIGHT. 
It is so called from Troyes, a city in France, where 
It was first employed ; it is used in philosophy, in 
weighing gold, &c. 

Graini » Symboh. 

OA •••... grs. 

^* • • niftke 1 pennyweight, 'wt. 

pennyweights 

480 or 20 . . i ounce, o%. 

I ounces 
6,760^ 240 or I 12 . 1 pound, . lb. 

A gram was originally the weight of a grain of corn, 
taken from the middle of the ear; a pennyweight, that of 
the Sliver penny formerly in use. 

APOTHECARIES WEIGHT. 
In mixing medicines, apothecaries use Troy weight, 
but subdivide it as follows : — 

Grains „ . , 

n(\ Symbolt 

^^ ' ' - ^ » make 1 sc< -ipk, ^1 



CO or 


scruples 
3 


• 


480 


24 or 

288 


drams 
8 


5,760 


90 or 



• • • 1 Uium, 6 

1 ounce, 5 
ounces 

12 . 1 pound, lb. 

its Carat," which is equal to four grains, is used in 

wo.iocMg di«monda. The term carat is alf.o applied in 

o^tiin:;nM>; Jie fineness of gold ; the latter, when ju.vf^nti,/ 



.♦ 



MBASU1RES. 



puro, *i8 said to bo " 24 oarata line." if there are 23 parta 
^oid, and one part gomo other material, the mixture is said 
to he "23 carats fine ; " if 22 parts out of the 24 are gold, 
it is " 22 carats fine," &c. ; — the whole mass is, in all caflos, 
supposed to be divided into 24 parts, of which the number 
consisting of j»old is specified. Our gold coin is 22 carats 
fine; puro gold being very soft would too soon wear out. 
The degree of fineness of gold articles is marked upon thera 
at I ho Goldsmith's Hall; thus wo generally perceive " 18" on 
the cases of gold watches; this indicates that they are " ly 
carats fine" — the lowest degree of purity which is stampcfd. 

JM. 
_ ^ 80 

An avoirdupoise ounce . 437^ 

A Troy pound . . 5,760 

An avoirdupoise pound . 7,000 
A Troy poimd is equal to 372- 9G5 French grammes. 

175 Troy pounds are equal to 144 avoirdupoise ; 
175 Troy are equal to 192 avoirdupoise ounces. 



CLOTH MEASURE. 



24 


• • 




nails 


9 or 


4 


36 


16 or 


27 


12 or 


45 


20 or 


54 


24 or 



make 1 nail. 



quarters 
4 
8 
5 
6 



1 quarter. 

1 yard. 

1 Flemish cU 
1 English ell. 
1 French clI 



Linas 
12 . 


». • 


144 or 


inches. 
12 


• • 


^432 


36 or 

198 
252 

7,920 
10,080 

8o!640 


feet 
3 


• 


2,376 
3,024 


lOior 
21 or 

660 
840 

5,280 
6.720 


yards 
7 


95,040 
120,960 


220 or 
280 or 

1,760 
2,210 


7r.().?.20 

1167^680 



LONG MEASURE. 
(It is used to measure Length.) 

«. make 1 inch. 



perches 

40 

40 



32 



on ^- 



20 



or 



. 1 foot. 


1 yard. 


1 English perch 
1 Irish perch. 


1 English fur long 
1 Irish furlong. 


a 1 Vn-liHl. '"!!« 



.i^ts^sLjLttrttx ticf 2'9h 



8 1 Irish railc. 



»? 




6 



MEASURED 






3 iiiclies 
3 palms 
18 inches 
6 f<^.et 
6 feet 
120 fathoms 



3 palm. 

1 span. 

1 cubit 

1 pace. 

1 fathom. 

1 cable's length. 



rnHpr rii'?/- ^ . . ^ ^"^^ a^e equal to 14 Endisli 
miiec. Ihe Pans foot is eaual tn 19-7Q0 v^ vl ."&"^^* 

the Roman foot to 11-604 liTih^ ^ u ^""^^'"^ '"^^««J 
luuL 10 ii OU4, and the French metre to 39383. 

, MEASURE OP SURFACES 
foot, a surface one foot long and one foot m^L', L.^ 



Square inches 
144 



1,996 

3&,304 
63,604 

^,668,160 
9,640,160 

6,272,640 
10,160,640 




ncake 1 «q. foot. 

1 square yard. 



10,890 
17,640 

{43,560 

70,560 



1 «q. En, 
1 sq.Iris 



porcli.' 
perah. 



4,014,489,600 27,878,400 
6,602,809.600i25. 158,400 



1 sq. Eng. rood. ' 
• «q. Irish rood. 

1 statute acre. 
1 plantation acre. 



3,097,600 102,400 
5,017,60oll02,400 



1 sq. En, 
1 sq. Iris 



mile, 
mile. 



rp, „ ' ■ • "'" 1 sq. irisn mile. 

crXfaff 'aoTmo^ 1'or" ^"-ejarda, and the Irish. 
12lVre Irish aero, ' l"" «1™'« Kngibi are equal to 

anl'The^lg' IKoS'-'^o'^T^'fJ'"'" ^"-, ^-^^ 
en„o] f,. 101 r-.:-!.'. ' ,' "^"^^ J^^nglish square milfis a^a 



MKASURES. 



MEASURE OP SOLIDS. 

^ The teacher will explain that a cube is a solid having 
six equal square surfaces; and will illustrate this by 
models or examples — the more familiar the better. A 
cubic inch is a solid, each of whoso a^x sides or faces is 
a sqitare inch ; a cubic foot a solid cooh of whose osi 
sides is a square fool ^ &c. 

Cubic inches 

I'^^^S . . , . . niftka 1 cub"? fot^vV 

I cubic feet 
27 . . . icubu5r»J4\ 



WINE MEASURE. 



Gills or naggina 



8 or 
32 

320 

576 

1,344 

2,016 

2,688 



pints 






') 






^ m % 


• 


• 




quarts 






8 or 


4 


• 


• 

3 


• 




gallon 




80 


40 or 


10 


« 


• 


144 


72 


18 


• 


« 


836 


168 


42 


. 


• 


504 


252 


63 


• 


• 


672 


336 


84 


• 

hogshe 


• 




Elds 


1,008 
2,016 


504 
1,008 


126 or 
252 


2 


• 


4 or 





xnai^^ i <«i».*. 

1 gaJoiX 

1 anker. 
1 runlet. 
1 tierce. 
1 hogsheaa 
1 punclieon 



pipes 

2 



4,032 

8,064 

in some places a gill is equal to half a pint. 
Foreign wines, &c., are often sold by measures differing 
from the above. 



1 pipe or butt 
1 tun. 



ALE MEASURE. 



Gallons 
8 



16 or 


firkins 

2 




82 


4 or 

6 

8 

12 


kilderkins 
2 


48 
64 
96 


8 or 
4 or 
6 or 



barrels 

u 



make 1 firkin. 

1 kilderkin. 

1 barrel. 

1 hogshead. 
1 puncheon. 
1 butt. 



MEASURES. 

BEER MEASURE. 




make 1 firkin. 
1 kilderkin 
1 barrel. 



1 hogshead. 
1 puncheon. 
1 butt. 



Pints 

4 or 

8 

16 

64 

152 
256 
576 



DRY MEASURE. 
(It is used for wheat, and other dry goods.) 



quarts 
2 



512 

2,048 
2,500 



4 or 

8 

32 

96 

12V. 

288 

2uG 

1,021 
1,280 



pottles 

2 



5,120 2,560 



4 or 

16 

48 
64 
144 

128 



gallons 
2 



512 
640 

1,280 



8 or 

24 

32 

72 

64 

256 
320 

640 



pecks 
4 



12or 
16or 
36 or 

32 

128 
160 



bushels 
3 
4 
9 



320 80 



8 or 

32 
40 



coombs 
2 



make 1 pottle. 

1 gallon. 

1 peck. 

1 bushel. 

1 sack. 
1 coomb. 
1 vat. 



8 or 
lOor 

20 



quarters 
4 
5 



1 quarter. 



1 chaldron 
1 wey. 
weys 
10 or I 2 Hast. 






riurds 
60 



TIME. 

MEASURE OE TIME. 



or 



3600, 

216,000 

5,184,000 



t 0,288,000 
. 15,152,000 
1,892,160,000 
1,897,344,000 

1,892,160,000 



seconds 
60 



3600 or 

86,400 

604,800 
2,419,200 
31,536,000 
31,622,400 

31,636,000 



minutes 
60 



1,440 or 

10,080 
40,320 
625,000 
627,040 

525,600 



hours 
24 



Ifi'^ 01 
672 01 
8,700 or 
8,784 01 

8,760 



Jays 
7 

28 
305 
366 

365 or 



SytiaoU 
make 1 second " 

1 minute « 

1 hour h. 

1 day d 

1 week w. 
1 lunar month. 
1 common year 
1 leap year. 

calendar mon.") 

12 I , 
lunar months f * y^^^' 

13 J 

The following Avill exemplify the use of the above symbols : — - 
The solar year consists of 365 d. 5 h. 48' 45" 30'": read » three 
hundred and sixty-five days, five hours, forty-eight minuteB. 
forty-five seconds, and thirty third;?. 

The number of days in each of the twelve calendar months 
will be easily remembered by means of the well known lines, 

"Thirty days hath September, 
April, June, and November, 
February twenty-eight alono 
And all the rest t)iirty-»ne." 

The follomng table vrill enable us to find how many days 
there are from any day in one month to any day in another. 



From any Day in j 


T. 

M 
>• 

Cl 

>• 

o 
H 

i 




Jan. 


Feb. 


Mar 


April 


May 


June 


July 


Aug. 


Sept. 


Oct. 


Nov 


1 

Dec 


Jan. 


36.5 


334 


306 


276 


245 


214 


184 


163 


122 


92 


61 


31 


Feb. 


31 


36.') 


337 


306 


270 
304 


246 


215 


184 


153 


123 


92 


62 
90 


Mar. 


59 


2S 


366 


334 


273 


243 


212 


131 


151 


120 


April 


90 


69 


31 


365 


.335 


304 


274 


243 


212 


182 


151 


121 


May 


120 


89 


61 


30 


365 


334 


304 


273 


242 


212 


181 


161 


June 
July 


151 
181 


120 


92 


61 


31 


365 


336 


304 


273 


243 


212 


182 


150 


122 


91 


61 
92 


30 
6) 


366 


334 


303 


273 


242 


212 
243 


Aug. 


212 


181 


153 


122 


31 


365 


334 


304 


273 


Sei)t. 


243 


212 


184 


153 


123 


92 


62 


31 


365 


335 


304 


274 


Oct. 


273 


242 


214 


183 


153 


122 


92 


61 


30 


366 


334 


304 
335 


Nov. 


304 


273 


245 


214 


1.4 


153 


123 


92 


61 


31 


365 


Dec. 


33^1 


303 275 


211 


214 


183 


163 


122 


91 


61 


30 


366 



I 



10 



TIME. 



iH plucod, and at the samJSf.' i *^^\^«^d of which March 
ihi left hand sSe of wWch ? On nT^ ^^' horizontal row at 
intersection the numW o]] • t '"' "^^ P"""^^^^ ^'^ ^^^^r 
tervene between ti el^fceenth 'oT M T clays therefore, in- 
October. But thrfourth nf o!. k '"' ' ^?^ *^^ fifteenth of 
than the fifteenth Zv?, 5/^ *"^T '" ^^^^«" d^y« earl-'er 
obtjdn 20i'S::r!^^^:^ ^^'^^^^^ " ^-- ^U, and 

^f^^^^^^ the 

bofore in tlie table, wo find tint Tin^ • ? Looking as 
th., third of J„„„,,y and tt th rd "ffc"t7rt^'"'-^''" 
tucntli IS sixteen days later thnn *!,„.;■ J' "'f' ""' "'"e- 
«nd obtaii, 136, the Jiit, required ■*' ™ ''^<' ^« '» ^^^ 

«Ud one to the 130, aXl'sV tuldtZlt^'r "^ "'"'"^ 

'w this bclZTnZl^lTtTl ^"- "■«J"«tand 
tbe Julian Calenda,;S !"«',,» T '" '."'""^ """ 
fourth year to the mojrirSarf tftM J P • ''^ 

Gregory, t.ro..d, this: „..dai™r«,s^;:;::;„„^r 



TIME 



11 



to the Julian style, would have been the 5th of October 
1582, should be considered as the 15th ; and to preveni' 
the recurrence of such a mistake, he desired that, in 
place of the last year of every century being, as hitherto, 
a leap year, only the last year of every fourth century 
should be deemed such. 

The " New Style," as it is called, was not introduced 
into England until 1752, when the error had become 
eleven days. The Gregorian Calendar itself is slightly 
inaccurate. 

To find if any given year be a leap year. Tf net the 
last year of a century : 

Rule. — Divide the number which represents the 
given year by 4, and if there be no remainder, it is a 
leap year. If there be a remainder, it expresses how 
long the given year is after the preceding leap year. 

Example 1.— 1840 waa a leap year, because 1840 divided 
by 4 leaves no remainder. 

Example 2. — 1722 was the second year after a leap year, 
because 1722 divided by 4 leaves 2 as remainder. 

If the given year be the last of a century :. 

ivULE.— Divide the number expressing the centuries 
by 4, and if there be no remainder, the given one is a leap 
year ; if there be a remainder, it indicates the number 
of centuries between the given and preceding last year 
of a century which was a leap year. 

Example 1. — 1600 was a leap year, because 10, being 
divided by 4, leaves nothing. 

Example 2.— 1800 was two centuries after that last year 
of a century which was a leap year, because, divided by 4, 
it leaves 2. • 



DIVISrW OF THE CIRCLE. 



Thirds 
60 



8600 or 

216,000 

77,760,000 



seconds 
60 



8,600 or 
1,296,000 



minutca 
60 



make 1 second " 
1 minute ' 
1 degree ° 



I degrees 
360 1 circumference. 
. Ev«»vy circle is supposed to be divided into the same 
P'^.'^^r of degrees, miautos, &c. ; the greater or less, there 



12 



DEFlNrnONf?. 







fore, the circle, the greater or less each of these will be. The 
following will exemplify the applications of the symbolej : — 
00° 6' 4" 6'" ; which means sixty degrees, five minutes, four 
sccondj, and six thirds. 



DEFINITIONS 



1. Arithmetic may be considered either as a science 
or as an art. As a science, it teaches the properties of 
numbers ; as an art, it enables us to apply this know- 
ledge to practical purposes ; the former may be called 
theore(tical, the latter practical arithmetic. 
' 2. J. Unit^ or as it is also called, Unity ^ is one of tho 
indivdduals nnder consideration, and may include many 
units of another kind or denomination ; thus a unit of 
the order called " tens" consists of ten simple units. Or 
it may consist of one or more parts of a unit of a higher 
denomination ; thus five units of the order of " tens" are 
five parts of one of the denomination called " hundreds ;" 
three units of the denomination called " tenths" are 
three parts of a unit, which wo shall presently term *the 
" unit of comparison." 

3. Numler is constituted of two or more units ; 
strictly speaking, therefore, unity itself cannot be con- 
sidered as a number. 

4. Abstract Nwmhers are those the properties of 
which are contemplated without reference to their appli- 
cation to any particular purpose — as five, seven, &c. ; 
abstraction l)eii^ a process of the mind, by which it sepa- 
rate^/ considers those qualities which cannot in reality 
exist by themselves ; thus, for example, when we attend 
only to the length of anything, we are said to abstract 
from its breadth, thickness, colour, &c., although these 
are necessarily found associated with it. There is nothing 
inaccurate in this abstraction, since, although length 
cannot exist without breadth, thickness, &c., it has pro- 
perties independent of them. In the samo way, five, seven, 
&c., can be considered only by an abstraction of the 
wind, as not applied to indicate soiae particuiar things. 

5. Ajjplicate Numbers are exactly the reverse of 



DICFINITIONS. 



13 



(rill be. The 
! symbolej : — 
aiuutes, four 



IS a science 

roperties of 

this know- 

y be called 

i one of tlio 
elude many 
i a unit of 
e units. Or 
of a higher 
" tens" are 
hundreds ;" 
;enth8" are 
ly term 'the 

lore units ; 
lot be con- 

aperties of 
their appli- 
even, &c. ; 
tich it sepa- 
t in reality 
Q we attend 

to abstract 
ough these 
e is nothing 
ugh length 

it has pro- 
, five, seven, 
tion of the 
xliii' things, 
reverse of 



abstract, being applied to indicate particular objects — 
as five men, six houses. 

6, The Unit of Comparison. In every number 
there is some unit or individual which is used as a 
standard : this we shall henceforward call -the " unit 
of comparison." It is by no means necessary that it 
should always be the same ; for at one time we may 
speak of four objects of one species, at another of four 
objects of another species, at a third, of four dozen, or 
four scores of objects ; in all these cases four is the 
number contemplated, though in each of them the idea 
conveyed to the mind is different — this difference arising 
from the different standard of comparison, or unity 
assumed. In the first case, the " unit of comparison" 
was a single object ; in the second, it was also a single 
object, but not of the same kind ; in the third, it became 
a dozen ; and in the fourth, a score of objects. Increas- 
ing the '' unit of comparison" evidently increases the 
(Quantity indicated by a given number ; while decreas- 
ing it has a contrary effect. It will be necessary to 
bear all tliis carefully in mind. 

7. Odd Numbers. One, and every succeeding alter- 
nate number, are termed odd ; thus, three, five, seven, &c. 

S. Evm Numbers. Two, and evcy succeeding alter- 
nate number, are said to be even ; thus, four, six, eight, 
&c. It is scarcely necessary to remark, that after taking 
away the odd numbers, all tliose which remain are even° 
and after taking away the even, all those which remain 
arc odd. 

We shall introduce many other definitions when treat- 
ing of those matters to which they felate. A clear 
idea of what is proposed for consideration is of tho 
greatest importance; this must be derived from tho 
definition by which it is explained. 

Since nothing assists both the understanding and the 
memory more than accurately dividing the subject of 
instruction, we shall take this opportunity of remarking 
to both teacher and pupil, that we attach much impor- 
tance to the divisions which in future shall actually be 
madr;, or shall be implied by the order in which tho 
different heads will be examined. 

b2 



M 



I t 



SECTION I. 

ON NOTATION AND NUMERATION. 

1. To avail ourselves of the properties of numbers, 
we must be able both to form an idea of them ouiselvcs, 
and to convey this idea to others by spoken and by written 
language ; — that is, by the voice, and by characters. 

The expression of number by characters, is called 
notation, the reading of these, numeration. Notation, 
therefore, and numeration, bear the same relation to 
each other as loriting and reading, and though often 
confounded, tliey are in reality perfectly distinct. 

2. It is obvious that, for the purposes of Arithmetic, 
we require the power of designating all possible num- 
bers ; it is equally obvious that we cannot give a dif- 
ferent name or character to each, as their variety is 
boundless. Wo must, therefore, by some means or 
another, make a limited system of words and signs 
Buffice to express an unlimited amount of numerical 
quantities : — ^with what beautiful simplicity and clear- 
ness this is effected, we shall better understand presently. 

3. Two modes of attaining such an object present 
themselves ; the one, that of comhining words or cha- 
racters already in use, to indicate new quantities ; the 
other, that of representing a variety of different quan- 
tities by a single word or character, the danger of 
mistake at the same time being prevented. The Romans 
Bimplified their system of notation by adopting the prin- 
ciple of combination ; but the still greater perfection of 
ours is due also to the expression of many numbers by 
the same character. 

4. It will be useful, and not at all difficult, to explain 
to the pupil the mode by which, as we may suppose, an 
idea of considerable numbers was originally acquu-ed, 
and of which, mdeed, although unconsciously, we still 
avail om-selves ; we shall see, at the same time, how 
methods of simplifying both numeration and notation 
Were naturally suggested. 



NOTATION AND NUMERATION. 



15 



of numbers, 
im ourselves, 
id by written 
aracters. 
rs, is called 
. Notation, 

relation to 
hough often 
;inct. 

Arithmetic, 
)ssible num- 
I give a dif- 
ir variety is 
e means or 
3 and signs 
if numerical 
^ and clear- 
id presently, 
ject present 
)rds or cha- 
mtities ; the 
Ferent quan- 
! danger of 
rhe Romans 
ing the prin- 
lerfection of 
numbers by 

t, to explain 
suppose, an 
ly acquu'ed, 
isly, we still 
3 time, how 
.nd notation 



I^.' lis suppose no system of numbors to bo as yo.i con- 
^(ruered and that a licap, for oxan.ph,, of pebbles, i.s 
p Mcod before us that wo may discover their amount. 
It this IS con.sulerablo, we cnimot ascertain it l)y look- 
ing at tlicui all together, nor even by separatdy in- 
specting them; we must, therefore, have recourse t(» 
that contnvnuce which the mind always uses when it 
desires to grasp what, taken as a whole, is too great for 
IS powers. It we exan.inc an extensive landscape, as 
the eye cannot take it all in at out view, we look sue- 
cessively at its different portions, and form our iud.^- 
n.ont upon them in detail. We must act similarly with 
retcrerice to large nunibers ; since we cannot compre- 
Jiend^thcm at a single glance, we must divide them into 
a suHiciout number of parts, and, examining these in 
succession, acquire an indirect, but accuratS idea of 
he entire. This process becomes by habit so rapid, 
that It seems, it carelessly observed, but one act, thou-h 
It is made up of nuiny : it is indispensable, whenc^cr wo 
clfsire to have a clmr idea of nunibeis— which is not 
Iiowcver, every time they are mentioned. ' 

5. Had we, then, to form for oursolVes a numerical 
sys eni, we would naturally divide the individuals to be 
reckoned into equal groups, each group consisting of 
S;mio number quite within the limit of our comprJlien- 
rum ; it the groups were few, our object would be attained 
vithout any further effort, .incc .ve should have acquired 
•urate kiiow edge of the number of groups, aiid of 
•n.l^er of individuals in each group, and therefore 
i ik.'tory, although mdu-ect estimate of the whole 

>.e^^ught to remark, that different persons have 
ery different Innits to their perfect comprehension of 
number; the mteliigent can conceive with ease a com- 

incapable of forn)mg an idea of one that is extremelv 
flmali. •' 

6. Let us call the nmnhr of individuals that we choose 

cons itute a group, the rafio ; it is evident that the 

larger the ratio, the smaller the number of groups ano 

the smaller the ratio <!.- Uyo-^^ fi.p -,,-,,.i- fe^""l''^> 'in^ 

&ut the smaller the number of groups the be-tter. 



16 



NOTAT[()?J AND NUMKRATION. 



7. If the groups into which wo havo divided tho 
objects to bo rockoueJ exceed in amount that number 
of whicli we havo a i)orf(!et idea, wo must continue tho 
process, and considorinj,' tho groups tlieniHolvos a.s indi- 
viduals, must form with them new groups of a higher 
order. Wc must thus proceed until tho number of^our 
highest group is sutFiciently .« ' ill. 

8. Tho raiio used for groups of the second and higher 
orders, would naturally, but not necessarily, be the samo 
as that adopted for the lowest ; that is, if seven indi- 
viduals constitute a group of tlie first order, we would 
probably make seven groups of the first order constitute 
a group of the second also ; and so on. 

y. It might, and very likely would happen, that wo 
should not have so many objects as would exactly form 
a certain* number of groups of the highest order — 
eomo of the next lower might be left. The same might 
occur in forming one or moi'e of the other groups. Wo 
might, for example, in reckoning a heap of pebbles, 
have two groups of the fourth order, three of the third, 
none of the second, five of the first, and seven indi- 
viduals or "units of comparison." 

10. If wo had made each of the first order of groups 
consist of ten pebbles, and each of the second oi-der 
consist of ton of the first, oacli group of the third of tm 
of the second, and so on with the rest, we had selected 
the deciiiuU system, or tliat which is not only used at 
present, but which was adopted by tho llcjbrew.s, Greeks, 
Komans, &g. It is remarkable that the language of 
every civilized nation gives names to the ' dilferent 
groups of this, but not to those of any other numerical 
system ; its very general diffusion, even among rude 
Hnd barbarous people, has most probably arisen from 
the habit of counting on the fingers, whidi is not 
altogether abandoned, even by us. 

11. It was not indispensable that we sliould havo 
ased the same ratio for the groups of all tho diiFereut 
orders ; we might, for example, have made four pebbles 
form a group of the first order, t^velve groups of the 
first order a group of the second, and twen.ty grouT)a 
of the second a group of the third order :— iu such a 






i 



(lividcd tho 
; that number 
b continue tho 
wives aH incli- 
is of a hii^lior 
lumber of our 

nd and higher 
r, be the same 
if seven indi- 
ler, we would 
der constitute 

ppcn, that wo 
exactly form 
jhcst order — ■ 
10 same mip;lit 
groups. \Vo 
p of pebbles, 

1 of the third, 
i seven indi- 

dor of groups 

second ordor 

e third of ton 

had selected 

only used at 

rews, Greeks, 

language of 

tho diiibrent 

ler numerical 

among rudo 

arisen from 

vhich is not 

should havo 

tho diiForent 

four pebbles 

•roups of the 

vonty o'voiiT)a 

:— iu such a 



NOTA'lloN AND NUATERATION. I7 

case we had adopted a system oxacllj Hko that to \m 

l.i. things make a group „f the order pma' twolv-o 
pcnr-o a g,H,up <,f the order di/fin^s, twenty shillms^ 
group of tho o.lor rour.ls. While ii urns "L ad LS 
that the use of tho su>ne system for anplicato ■ s fV, r 

p^ ^:"e "';;f 'vv^'' s-utiy si,..piity ^r :;is;."ti^ 

?iT •''■' H' ''''y ^'''^'-^"^ hereafter, a dance 
^}t the tab OS g,v.m already, and those set down in t^^o' tt- 
ag of exciyjugo, will .how that a great vanoZ^X^ 
have actu;illy been consti-uctcd. «>> stems 

12. When wo use the same ra/io for tho groups of all 
h.. orders, we term it a cnmuon ratio. The,^ aX' s 
I'-tvo boon nr> particular reason why /...slumid So Lee. 
seloctod as a ' connuon ratio" i„ the systeuM/f numbe' 

.a.,1 od, by the mode of counting on the fimrors • a„,l 
that It is neither so low as unnecessarily Si .Ve 

'0 number 01 orders of groups, nor so higlf afto oxc^^i 
J^o ...jceptmn of any one ibr whom the sys;;^^' 

. ■'"!; ^^;^y«^^^'" i'l ^^hi«li ten is the "common ntio^' 

r.n — ouis IS, taoroforo, a " docimal .'system" of numbe.s 
It the common ratio were sixty, it would b-^ a y ' . • " 

^^^: :;;;'7.^"^^- i^-ony usod,;ndt^m 
-i e ft. 'L T P^^'^'-^ived by the tables already 

g.vcn toi the measurement of arcs and an-los and of 
time. A ^n,na,y .system wOuld have five ^ r k s' - com 
mon ratio - a cluoaecnal, twelve ; a vl^.i^^^l: tv.^nt^; 

14. A little reflection will show that it was use-loss 
to give diflorent ikumos and chT-.,pfp,.« f/ "■''^^loss 

except to those which a "lo^'h ^ " hnt^'ILw^'l" 
tutos the lowest g.-oup, and to th #.:; 1 ^ilf 

ss.;.d^r:v^:^ '-'''''' nun;boi.iast'c:;:^;: 

oviaent-4i-b:u;:;;^h;;;r^t:;r^^^^^ 



18 



NOTATION AND NUMERATION. 



Nnnttt, 




Chitntetvt. 


One 




1 


Two 




9 


Til 100 




.<} 


l''oiir 




4 


Kivo 




A 


Six . 




6 


Se'en 




7 


Kiiht . 




8 


Nu.o 




SI 


Ten 




10 


Hundred 




luo 


'J'lioiisund 




1,000 


'J'eii tliotiKn 


nd" 


lo.wm 


lliiiuliud til 


Dusa 


nd ico.ooo 


Million . 


. 


1,000,000 



is just what wo liiivo dnno in our numnrioal py.stoni, 
excofit tliat we have rormcd tho nnnioH of koiuo of tlio 
groups )»y cmubination (»f tlioso nlrcndy used ; thus, 
*' tens of tliouvinTi(i.M," the grou]) n;'xt IihtIkm' th.-in lliou- 
sandn, is designated by a conibinntioii of words ah-cady 
ajtpliod to express other groups — whieli tends yet further 
to siniplilication. 

15, ARADIC SySTEJM OF NOTATION : 



UdUs ol uonijiai'lson, 



First grotip, or units of tho second order, 
Seooiiii ({roup, or iiiiitti of the third ordt-.r, 
Tliinl Kronp, or units of tho fourth order, 
Fourth groii]), or units of the llfth order, 
l''ilih f;r()ii|), or units of the sixth order, 
Sixth gronii, or units of tho .seventh order, 

Ifi. The characters whicli express tlio nine first nuTu- 
bers are tho only ones used ; they are called (/i.gi/s^ JVoin 
tho custom of counting thoni on the lingers, already 
noticed — " digitus" meaning in Latin a finger ; Ihay aie 
also called significant figures, to distinguish them tV.,-m 
the cypher, or 0, which is used merely to give the digits 
their proper 'position with ret'crcncc to tht; dcciinal point. 
The pupil will distinctly remember that the place where 
the "• units of comparison" are to be found is that imme- 
diately to the left hand of this point, which, if n(»t ex- 
pressed, is supposed to stand to the right hand i>id(.; of 
all tho digits — thus, in 4(JS-7G tho 8 expresses " units 
of comparison," being to the left of the decimal point ; 
in 40 the 9 expresses " units of comparison,'"' the deci- 
mal point being understood to tlie riirlit of it. 

17. We find by the table just given, that after the 
nine first numbers, the same digit is constantly repeaieil, 
its position with reference to tho di-ciaal point being, 
however, changed : — that is, to indicate each succeeding 
group it is moved, by means of a cyplier, one place 
farther to the loft. Any uf the dibits may be uaed to 



N. 

fiorionl pystoni, 
of 8omo of (lio 
y usod ; thus, 
rlier tlinn tliou- 
wotds nlrcjuiy 
!U(Is yot further 



Chantetvt. 
1 

a 

4 

A 

6 

7 

8 

SI 

10 

10(1 

I,(J()() 

10,(HI0 



NOTATION AND Nt/MERATIOrf 



19 



!<1 

ind 
niKnnd 



d thousand l(:0,(il)(i 
l,()00,f)(;o 

nine fii-Ht miTii- 
iod ili.gih^ JVoiii 
iiigor.s, already 
iigcr ; Ihay are 
ish thciiu fr.,"in 

give tlie di^i<s 
! decimal p. int. 
he place where 
d \H that imtue- 
ieh, if nut ex- 
t hand side of 
presses " uuitH 
decimal point ; 
son,'' the deci- 
■ it. 

that after tlse 
antly repeated, 
d i)oint being, 
ach succeeding' 
lier, one place 
ay be used to 



flxpreHs its respectiv(3 n.iniber „f any of the .rroups •-. 
thus S would be eight u ...j,, ./.nnpari^onTH^ 
oi-ht groups ot tho first ord<n-, or ei-dit " tens" of 
Mmpic units ; 800, eight groups of tho .second, or' unit^ 
f tho h.rd order ; and so on. Wo might use any of 
tho digits With tho different groups ; thus, for examDio 

7 ;;;; r^'n^^v'^'^ "^i"^^ 3?o;those'of;he " S; 

•Z'to r *'^:: ^^^1^''" •?^'*^ J™ i'l f>ill would be 5000, 
3 0, 70 8, or for brevity sake, 5378-for wo never nm 
tho cypher when we can supply its place by a si^^nificant 
ftguro, and it is evident that in 5378 t!ie 378 ko p^^^^^^^ 
T) tour p.aees from th. decimal point (understood), fult aa 
wo as cyphers would have done ; a\so the 78 k^ee'pT ho 
3 Y the third, and the 8 keeps the 7 in tho second placo. 
lb. It IS important to remember that each di^rit hag 
two values, au absoUta and a relative; tho absolute 

un " m v' h ""'"i '' ""^^^^ '^^'^'^'^^ whatever tte 
units may bo, and is unchangeable: thus 6 alwavs 
I'/oans SIX, sometimes, indeed, Sx tens, at otlu^r tSa 
Hix hundred, &c. The relative value depends on the 
o dcr of units mdicated, and on the nature if tho "unit 
ot comparison." * 

_ la. What has been said on this very important snli- 

l;;ct, IS intended pi-incipally for the teLheir both t 

o.-dmary amoun of industry and intelligence wil be 

a ci ld,_ paitieulariy it each point is illustrated by an 
appropriate example ; the pupil may be made f\ f in 
.stance, to arrange a numbe/of pebbles in groups son "- 
ti.nes of one, sometimes of another, and soiZinies of 

S'od f 1 ;n7 • 1 1 '°'"P^'''^'' ^^'^"^o^ occasionally 
,, Changed tiom individuals, suppose to tens, or hundreds op 
\ scores, or dozens, &c. Indeed the pupils J fbe well 
; oquamted witli these introductory ma^tters otherw^o 

., verj acnnite ideas of many th n'>-s thev will be poIl/i 

inijc£ii.y to understand. A.n" t"0'ihia Un •f-.ixr i i ,^ 
toaoho. at thi. „.,„,, ,,ni f,o we,f;o,«i;n;'ttt 



(i 






■ 



111 



20 



NOTATION AND NrMliRA'I'IOV. 



and rapidUj^ with Aviiicli tlic sdiolar will cafterwardg 
adviincjo ; to be assured of this, ho has only to recol- 
lect that most of his future reasonings will be derived 
from, and his explanations grounded "on the very prin- 
ciples we have endeavoured to unfold. It may be taken 
as an important truth, that what a child learns without 
undcratanding, he will acquire with disgust, and will 
uoon cease to remember ; for it is with children as with 
persons of more advanced years, when wo appeal suc- 
cessfully to their understanding, the pride and pleasure 
they feel hi the attainment of knowledge, cause the 
labour and the weariness which it costs to be under- 
valued, or forgotten. 

20. Pebbles will answer well for examples ; indeed, 
their use in computing has given rise to the term calcu- 
lation., " calculus" being, in Latin, a pebble : but while 
the teacher illustrates what he says by groups of par- 
ticular objects, he must take care to notice that hi^^ 
remarks would be equally true of any others. He must 
also point out the difference between a group and its 
e(juivak'nt unit, which, from their perfect equality, are 
generally confounded. Thus ho may show, that a penny, 
while C(iual to, is not identical with four farthings. This 
seemingly unimportant remark will be better appre- 
ciated hereafter ; at the same time, without inaccuracy 
of result, we may, if we please, consider any group 
dther as a unit of the order to wliieh it 1)clongs, or so 
many of the next lower as are equivalent. 

21. lloman Notaiion. — Our ordinary numerical cha- 
racters have not been always, nor every where used tc 
express numbers ; the letters of the alphabet naturally 
pi-esonted themselves for the pui-pose, as being already 
familiar, and, accordingly, were very generally adopted— 
for example, by the Hebrews, Greeks, llomans, &c., 
each, of course, using their own alphabet. The pupil 
should be acquainted with the lloman notation on 
account of its beautiful simplicity, and its being still 
employed in inscriptions, &c. : it is found in the follow- 
ing table : — 



ill afterwardg 
only to rccol' 
dll be derived 
:lio very prin- 
may bo taken 
learns without 
;nst, and will 
ildren as with 
e appeal suc- 

and pleasure 
50, cause the 

to be under- 

ples ; indeed, 
le term calmt- 
le : but while 
roups of par- 
tice that \m 
rs. He must 
;roup and its 
equality, are 
that a penny, 
'things. This 
better appre- 
Lit inaccuracy 
r any group 
)elongs, or so 

imerical cha- 
vhere used tc 
.bet naturally 
being already 
lly adojited — ■ 
iomans, &c., 
u The pupil 
notation on 
ts being still 
in the follow- 






NOTATIOi'T AND NUMERATION. 



ROMAN NOTATION. 



31 



Characters. 
I. 

n. 
in. . 

Anticipated change IIII, or IV. 
OJiauge . . V. . 

VI. . 

VII. . 

VIII. . 
Auticipaletl change IX. 



Change 



X. 
XI. 
XII. 
XIII. 

xtv. 

XV. 
XVI. 
XVII. 
XVIII 

xrx. 

XX. 



A uticipated change X [, 
Change . . L. 

. . T'X., &c. 

Anticipated change XO. 
Change . . C. 

, . ". CO., &c. 

Anticipated change CD. 
Cliango . . D. orr>. 

Anticipated change CM 
Ciiauge , ' . U. or CIq 

y. or 1,30 



&c 



JSiimhers Exprrsstd. 

. One. 
. Two. 
. TJiree 
. Four. 
. Five. 
. Six. 
. Seven. 
. Eight. 
. Nine. 
. Ten. 
. Eleven. 
. Twelve. 
. Thirteen. 
. Fourteen. 

Fifteen. 

Sixteen. 

Seventeen. 

Eighteen. 

Nineteen. 

Twenty. 

Thirty, &c. 

Forty. 

Fifty. 

Sixty, &c. 

Ninety. 

One hundred. 

Two hundred, kc 

Four hundred. 

Five hundred, &c 

Nine hundred. 

One thousand, &c. 



Five thousand, &c. 
X. or CCIoo • Ten thousand, &o. 
laop. . Fifty thousand, &cj. 

'^'-'CiOOO. • One liundred thous.and, &c 
. 2,2. Thus we find that the liomuns used vnrv fe^7 
eiiaracters-fower, iridoed, than we do, althou-Wl our 
sjstoMi ,s stdl more snaplo and eiToctive, from our ap,>K-- 
ing.tli. prmc.pl. of "position," unkuo-.vn to them ^ 
llicy expressed all numbers by the followiiv. symbol. 
or combmatiou.3 of them : I. Y. X. L. (J. D. ov U T" 
orCLo.^ In cojistrueting their system, they evidently 
had a (pmary m view ; that is, as we have ^rid, an. I 
wlucii nve would be the common rallo ; for we find that 
luey ciians'tiu th 



heir eharactur, not only at ten, toij t 



iine» 



i i 




in. 



22 



WOTAT ^,^ Aww wwMEKATION. 



ten, &c., but also at five, ten times five, &c. : — a purely 
decimal system would suggest a change only at ten, ten 
times ten, &c. ; a purely quinary, only at five, five times 
five, &c. As far as notation was concerned, what they 
adopted was neither a decimal nor a quinary system, 
nor even a combination of both ; they appear to have 
supposed two primary groups, one of five, the other of 
ten " units of comparison ; " and to have formed all the 
other groups from these, by using ten as the common 
ratio of each resulting series. 

23. They anticipated a change of character; one 
unit before it would naturally occur — that is, not one 
" unit of comparison," but one of the units under consi- 
deration. In this point of view, four is one unit before 
five ; forty, one unit before fifty— tens being now the 
units under consideration ; four hundred, one unit before 
five hundred— hundi-eds having become the units con- 
templated. 

24. When a lower character is placed before a 
higher its value is to be subtracted from, when placed 
after it, to be added to the value of the higher ; thus, 
IV. means one less than five, or four ; VI., one more 
than five, or six. 

2b. To express a number by the Roman method of 
notation : — 

Rule.— Find the- highest number within the given 
one, that is expressed by a single character, or the 
" anticipation " of one [21] ; set down that character, 
or anticipation— as the case may be, and take its value 
from the given number. Find what highest number 
less than the remainder is expressed by a single charac- 
ter, or " anticipation ; " put that character or " anticipa- • 
tion "^ to the right hand of what is already wi'itten, and 
take its value from the last remainder :' proceed thus 
until nothing is left. 

Example.— Set down the present year, eighteen h'mdred 
and forty-four, in Roman characters. One thousand, ex- 
pressed by M., is the highest number within the giynn one, 
indicated by one character, or by an anticipation; we pu k down 

and take one thousand from the given number, which loaves 



^S 






r. 

ic. : — a purely 
tily at ten, ten 
EJve, five times 
ed, what they 
linary system, 
ppear to have 
, the other of 
formed all the 
LS the common 

laracter ; one 
it is, not one 
3 under consi- 
ne unit before 
eing now the 
ne unit before 
he units con- 

jed before a 

when placed 

ligher ; thus, 

I., one more 

n method of 

lin the given 
icter, or the 
at character, 
take its value 
^hest number 
lingle charac- 
or " anticipa- • 
written, and 
proceed thus 



teen h'lndred 
thousap.d, ex- 
the givon one, 
; we pu k down 

which loaves 



NOTATIOM AND NCTJIERATION. 



23 



'IS' 



eight hundred and forty-four. Five hundro „ tho 

highest number within tho last remainder (e /./^ fu.r,dred 
and iory-four) expressed by one character, or jin "antici- 
pation ;■' we set down D to the rin;ht liand of M 

JNID, 
and take its value from eight hundred and forty-four, which 
leaves three hundred and forty-four. In this^the hio-host 
number expressed by a single character, or an "anticipa- 
tion, ' IS one hundred, indicated by C ; which we set down ; 
and tor the same reason two other Cs 
Tw , MDCCC. 

This leaves only forty-four, tho highest number withic 
which, expressed by a single character, or an "anticipation '' 
IS torty, AL— an anticipation ; we set this down alsof 
_ MDCCCXL. 

Four expressed by IV., still remains; which, being als« 
added, the whole is as follows:— « uj, ms* 

MDCCCXLTV. 

26 Posiiion.—The samp .jharacter may have dif- 
terent values, according to tie place it holds with refer- 
ence to the decimal point, or, perhaps, more strictly, 
osiiion comparison." This is the principle of 

27. The places occupied by the units of the different 
orders, according to the Arabic, or ordinary notation 
[lo] , may be described as follows :-units of comparison, 
one place to the left of the decimal point, expressed 
01- understood ; tens, two places ; hundreds, three places 

tl K 1 F""^'^ '^'''"^•^ ^' "^^'^^ '^'^ ^^™ili^r with these 
as to be able, at once, to name the « place" of any order 
of units, or the " units" of any place ^ 

28 When therefore, we are desh-ed to write anv 
number, we have merely to put down the digits expres- 
sing the amounts of the different units in their prler 
places, according to the order to which each belongs, 
it, in the given number, there is any order of which 

domi u the place belonging to it ; the object of which 
2r^ ^7 'I::;, "Snifica. t figures in^ thpir own posi- 



mns. A cypher produces no effect wh 
between significant figu^^s and the 
0536, 636-0, and 536 



on it is not 

decimal point; thus 

would mean the same thing — the 



24 



NOTATION AND NUMERATION. 



second s, however, the correct form. 536 and 5360 dre 
different ; in the latter case the cypher affects the value 
because it alters the position of the digits. ' 

Example —Let it be re uired to set down si^ hundred 
h' ; ' Sr "^*^,"«' ^^2 : without the cypher, the six would 

29. In numerating, we begin with the digits of the 
highest order and proceed downwards, stating the num- 
ber which belongs to each order, fe nuiu 

clJhV'tf'^''^'' notation and numeration, it is usual to 
divide the places occupied by the different ordei of 
units into periods ; for a certain distance the Englisn and 
1 rench methods of division agree ; the English bHlion 
IS, however, a thousand times greater than the French 
Ihis discrepancy is not of much importance, since we 
aie rarely obkged to use so high a number,-we shall 
prefer the IWh method. tS give some idea of the 
amount of a billion, it is only necessary to remark that 
according to the English Method ot' notation 'there 
lias not been one billion of seconds sin e the birth of 
thrist. Indeed, to reckon even a million, counting on 
an average^ three per second for eight hours a day, 
would require nearly 12 days. The following are the 
two methods. ° 



\* 



Jill 



ENGLISH METHOD. 

Twin nnn- B^^ions. Millions. 

000-000 000-000 000-000 



Units. 
000-000 



Billions. 

Quiidivila. 'J'ens. Unita. 





FRENCH METHOD. 

Millions. Thousands. Units. 

Himd. .ens. Units. H„nd. Ten.. Unita. Hnnd, Ton.. Unit* 

000 000 000 

so Use of Periods.— Let it be required to read off 
the following number, 576934. We put the first point 
to the left of the hundreds' place, and find that there are 
exactly two periods— 576,934 ; this does not always 
occur, as the higlicst period is often imperfect, consisting 
oniy 01 one or two digits. Dividing "the number thus 



NOTATION AND NUMERATION. 



and 5360 jlre 
cts the value, 

1 six hundred 
the two in the 
^her between 
the six wouhi 
mean not six 

digits of the 
ng the num- 

t is usual to 
it ordei of 

Englisii and 
iglish billion 
the French, 
ee, since we 
p, — ^we shall 

idea of the 
•emark, that 
ation, there 
he birth of 
counting on 
ours a (lay, 
ing are the 



Units. 
000-000 



Units. 

nnd. Tpi)«. Unit* 



to read off 
i first point 
it there are 
not always 
, consisting 
imber thus 



2b 



into part,^, shows at once that 5 is in the third place of 
the second period, and of course in the sixth place to 
the 4eft hand of the decimal point (understood) ; and, 
therefore, that it expresses hundreds of thousands. The 
7 being in the fifch place, indicates tons of tliousands ; 
the tJ in the fourth, thousands ; the 9 in the third, hun- 
dreds ; the 3 in the second, tens ; and the 4 in the first, 
units (of " comparison ") . The whole, therefore, is five 
hundreds of thousands, seven tens of thousands, six 
thousands, nine hundreds, three tens, and four units,— 
or more briefly, five hundred and seventy-six thousand, 
nine hundred and thirty-four, 

31. To prevent the separating point, or that which 
divides into periods, from being mistaken for the decimal 
point, the former should be a comma (,)— the latter a 
tuli stop (•) Without this distinction, two numbers 
Y.o^'i^f'' ^?^ different might be confounded : thus, 
498- /63 and 498,763,-one of which is a thousand 
times greater than the other. After a while, we may 
dispense witli the separating point, thouo-h it is conve- 
nient to use It with considerable number, as they are 
tiien read with greater case. ♦ 

32. It will _ facilitate the reading of large numbers 
not separated mto periods, if we begin with the units of 
comparison, and proceed onwards to the left, saymg at 
the fi,« digit ;' units," at the second " tens," at the 
tlnrd 'hundreds," &c., marking in our mind the deno- 
mination of i\iQ highest digit, or that at which we stop. 
We then commence with the highest, express its number 
and denommation, and proceed in the same way with 
each, until we come to the last to the right hand. ■ 

.+ !;^'-'^^^P;'J=— J^et it be required to read off 6402. Lookins 

ti t;^-'tlT"'^VV'^^r ^^y """^*^' ' '^' theO, ''tens;i 
?,fv^ fi ' V^^"^,^*«.<i«i" .a^d at tlie C, -thousands.- The 
.it.„er, therefore, being six thousands, the next di-it is f(,nr 
hundrod., &o Consequently, six thousands, fourlunu r ds 
lV^ rf two units; or, briefly, six tliousand four hun 
drcd and two, is the reading of tlie given number. 

"m" ^^'t^ "'''^ ^^° ^•^^'^ *'^ facilitate notation, 'i^he 
pupil will fu-st write domi a number of Doiit^ds of cyphers 



26 



NOTATION AND NUMERATION. 



to roprosenf, tlio places to be occupied by the varioua 
orders of imit.s. Jle will then put the digits express- 
ing the diflerent denominations of the given number 
under, or instead of those cyphers which are in corres- 
ponding positions, with reference to the decimal points 
bogmumg with the highest. 

ExAMPLE.-Write down three t'nousand rlx hundred and 
btty-tour The highest dent ■< • -^a being thousands, will 
occupy the fourth place to thr ' ' the deoimal point. It 
will be enough, therefore, to ^r down four cyphers, and 
under them the corresponding digits-that expressing the 
thousands under the fourth cypher, the himdreis under tho 
third, the tens under tho second, and the units under the 
hrst; thus 

0,000 
3,654 
A cypher is to be placed under any denomination iu 
which there is no significant figure. 

Example.— Set do^vn five hundred and seven thousand, 
and sixty-three. ' 

000,000 
507,063 

After a little practice the periods of cyphers will 
become unnecessary, and the number may be rapidly 
put down at once. 

34. The units of comparison are, as we have said, 
always found in the first place to the left of the 
decimal point ; the digits to thr left hand progressively 
increase in a tenfold degree—those occupying the first 
place to the left of the units of comparison being tea 
times greater than the units of comparison ; those occu- 
pying the second place, ten times greater than those 
which occupy the first, and one hundred times greater 
than tho units of comparison themselves ; and so on. 
Moving a digit one place to the left multiplies it by 
ten, that is, makes it ten times greater ; moving it two 
places multiplies it by one hundred, or makes it one 
hundred times greater ; and sc of the res^ If all the 
digits of a quantity be moved one, two, &c., places to 
the left, the whole is increased ten, one hundred, &c., 
times— as tho case may be. On the other hand, moving 



i 



NOTATION AND NUMERATION. 



87 



' the various 
'^its express- 
fen number, 
re iu corres- 
uial point-— 

hundred and 
ousands, will 
lal point. It 
3yphora, and 
f pressing the 
ids under tho 
ts under the 



mination ia 
sn thousand, 



yphers will 
be rapidly 

have said, 
eft of the 
'ogressively 
ig the first 
. being ten 
those occu- 
than those 
ics greater 
md so on. 
plies it by 
i^ing it two 
kes it one 

If all the 
, places to 
idred, &c., 
ad, moving 



a digit, or a quantity one place to the right, divides ifc. 
by ten, that is, makes it ten tim^-s smaller than before ; 
moving it two places, divides it by one hundred, or 
niiikes it one hundred times smaller, &c. 

So. We possess this power of easily increasing, or 
diminishing any number in a tenfold, &c. degree, whetlicr 
the digits are all at the right, or all at the loft of the 
docimal point ; or partly at the right, and partly at the 
loft. Though we have not hitherto considered quautitiea 
to tho left of the decimal point, their relative value will be 
very easily understood from what wo have already said. 
For the pupil is now aware that in the decimal system 
the quantities increase in a tenfold dej^ree to the loft, 
and decrease iu the same degree to "the right ; but 
there is nothing to prevent this decrease to "the right 
from proceeding beyond the units of comparison, and 
tho decimal point ;— on the contrary, fi-oni the very 
nature of notation, we ought to put quantities ten times 
loss than units of comparison one pl-ice to the right of 
them, just as we put those which are ten times less than 
hundreds, &c., one place to the right of hundreds, &c 
We accordingly do this, and so continue the notation 
not only upwards, but downwards, calling quantities U 
the left of the decimal point integers, because none of 
them is loss than a whole " unit of comparison :" an^ 
those to the right of it decimals. When there are deci- 
mals m a given number, the decimal point is actuallv 
expressed, and is always found at the ri.<(ht hand side 
ot the units of comparison. 

30. The quantities equally distant from tho unit of 
comparison bear a very close relation to each other 
which IS indicated even by the similarity of their names • 
those which are one place to the left of the units of com- 
parison are called " tens," being each identical with -«r 
equivalent to ten units of comparison; those which are 
one place to the riffht of the units of comparison tim 
called tenths," each being the tenth part of, tlint is, ten 
times as small as a unit of comparison ; quantities two 
places to the Ipfi of the units of comparison are called 
imndrcds" being one hundred -times greater ; and 
those two places to the rigAl, " hundredths," beincr one 



S8 



h )••; 



■M 



NOTATION AND NUMERATION 



hun red times loss t u.n tlio units of comparison ; and .a 
ot .11 11.0 o hers to tl.c right and loft. This will bo mo o 
evident on in.sp.,cting the following tabic :-- 



Asrcirling Scries, or Integers 
Uiic Liiit 

Ilumlred . . jqq 

Thousand . . 3,000 

Ion tliou,s!ui(.l . 10,000 
Hundred tlioiisand 100^000 
&c. 



II' 



IVsccmlin-.' Sories, or DccliiMls. 
Ouo Unit. 
Tenth. 
Htindrcdtli. 
Tiioijsiindth. 
Ten-tliousfindth. 
Hundred- thousandth. 

&G. 



•1 

•01 

■001, 

•000,1 

•000,01 



We have seen that when we divide integers into periods 
Ifih.^ .separating point must be put to the rid.t 

of the thousands; m dividing decimals; the first poin 
nius^ be put to the right of the thousandths. ^ 

37. Oare must be taken not to confound what we 
Tl "ul !^''^'^': ^v'itli what wo shall hereafter des I! 

not irlent ica lly the same quantities-the decimals beino- 
wha sha 1 bo termed the " quotients" of the con-e.^ 
pondmg decimal ft.actious. This remark is made hero ^ 

i^n''? %T'"''''"'^^" ''^'^ ^^^ ^^' subject, in those 
who already know something of Arithmetic 

T'x. '^^"^l^ ^'^ ^'^ ^■'^''^"^^ ^"^'' treating integers and deci- 
mals bj dfent rules, and at ditForc;^t tin!es, since the^ 
follow precisely the sa.ne laws, and constitute parts of 
Uie very same series of numbera ]]esides, any quantity 
nay, as far as the decimal point is concerned, be ex- 
pressod m diflerent ways; tbr this purpose ^e ha;e 
Jierely to change the unit of comparison. Thus let it 
be required to set down a number indicating five hun- 
dred and seventy-four men. If the " unit of compari- 
son ho one man, the quantity would stand as follows, 
&74. If a band of ten men, it would become 57-4— f^r 
as each man would then constitute only the tenth pa,J 
of the "unit of comparison," four men would be only 
our-tenths, or 0-4 ; and, since ten men would forn. bu^ 
one unit, seventy men would be merely seven uni^s of 
comparison, or 7 ; &c. Again, if it wen. a band of one 
ImM mm, the number must be ^vritten 5-74 ; an-] 
I'lbiij, ir a miiKi ol a Inuusand 9mij it would be 0-574 



m 



NOTATION AND NUMERATION. 



29 



son ; and so 
will be more 



, or DooJiiwls. 

:h. 

Ith. 

siindth. 

•tliousfindth. 

Into periods 

to the rJLdit 

first point 

d what we 
after dosici;- 
equal, but 
inials beiniT 
the corres- 
;ide here to 
t, in those 

and deci- 
sinco they 
3 parts of 
y (juantity 
ed, be ex- 
' wo have 
'hus, let it 
'l five him- 

conipari- 
ts follows, 
37-4— for, 
enth pact 
1 bo only 

form but 
I units (»f 
lid of one 
■74 ; and 
be 0-574 



Should the " unit " bo a band of a dozen, or a score 
men, the change would be still more complicated ; as, 
not only the position of the dechual point, but the very 
digits also, would be altered. 

39. It is not necessary to remark, that moving the 
decimal point so many places to the left, or the digits 
an equal number of plaees to the right, amount to the 
same thing. 

Sometimes, in changing the decimal point, one or 
more cyphers are to be added ; thus, when we move 42 '(3 
three places to the left, it becomes 42600 ; when wo 
move 27 five places to the right, it is '00027, &c. 

40. It follows, from what we have said, that a deci- 
mal, though less than what constitutes the unit of com- 
parison, may itself consist of not only one, but several 
individuals. Of course it will often be necessary to indi- 
cate the " unit of comparison,"— as 3 scores, 5 dozen, 6 
men, 7 companies, 8 regiments, &c. ; but its nature does 
not affect the abstract properties of numbers ; for it is 
true to say that seven and five, when added together, 
make twelve, whatever the unit of comparison may be :— 
provided, however, that the sa7?ie standard be applied to 
both ; thus 7 men and 5 men are 12 men ; but 7 men 
and 5 horses are neither 12 men nor 12 horses.; 7 men 
and 5 dozen men are neither 12 men nor 12 dozen men. 
When, therefore, numbers are compared, &c., they must 
have the same unit of comparison, or — without alterin^r 
their value — they must be reduced to those which have"? 
Thus we may consider 5 lens of men to become 50 
individual men— the unit of comparison being altered 
from ten men to om man, without the value of tho 
quantity being changed. This principle must be kept 
m mind from the very commencement, but its utility 
will become more obvious hereafter. 



EXAMPLES IN NUMERATION AND NOTATION. 

JVoiaiion. 

1. Put down one hundred and four 

2. One tliousand two hundred and forty 

3. Twenty thousand, three hundred and forty-five 



^iu.i. 



104 

1,240 

20,345 



^ 



i ! 



mV. 



: t 



i' 



i 




•so 



NOTATION AND NUMERATION. 



5. 



G. 
7. 



Two Imndrod and tliirty-fuur thousand, fivo 
hundred and sixty-seven 
Three hundred and twenty-nirio tliousand, 
seven hundred and sevcnly-nine 
Seven hun(h-ed and nine tliousand, eight hun- 
dred and twelve . . *! . 
Twelve hundred and fo/ty-soveu tliousand, 
four hundred and tiCty-sovon 

8. One million, three hundred and ninety-seven 
thousand, four hundred and seventy-live 

0. Put down fifty-four, seven-tenths 

10. Ninety-one, fivo hundredths . 

11. Two, three-tenths, four thousandths, and four 
hundred-thousandths 

12. Nino thousandths, and three hund]-ed thou- 
sandths • • . . . 

13. Make 437 ten thousand times greater 

14. Mako 2 7 one hundred times greater 

15. Mako 0056 ten times greater . . 
10. Make 430 ten times less 

17. Mako 2-75 one thousand times les^i . 



Jln.i. 

234,507 

320,771) 

709,812 

1,217,457 

1,397,475 
54-7 • 
0105 

2*30401 

000903 

4,370,000 

270 

0-56 

43 

000275 



Numeration 



7. read 8540320 



5210007 
Gi.)30405 
50- 0075 
3' 000000 
00040007 



2. — 407 8. 

3. — 2700 • 0. 

4. — 5000 10 
6. — 37054 n. 
G. — 8700002 12. 

13. Smnd travels at the rate of ahout 1142 feet in a 
seeond ; light moves ahout 195,000 miles in the same time. 

14. The sun is estimated to be 880,149 miles in diameter: 
its size is 1 377,013 times greater than that of the earth. 

15. Tho diameter of the planet mcreurv is 3,108 miles, 
and his distance from the sun 30,^; 14,721 miles. 

10. The diameter of Venus is 7,498 miles, and her dis- 
tance from the sun 08,791,752 miles. 

_17. The diameter of the earth is a])out 7,904 miles: it is 
95,000,000 miles from tlie sun. and travels round the latter 
at the rate of upwards of 08,000 miles an hour. 

18. The diameter of the moon is 2,144 miles, and her dis- 
tance from the earth 230,847 )niles. 

10. The diameter- of Mars is 4,218 miles, and his distance 
from the sun 144,907,030 miles. 

20. The diameter of Jupiter is 89,009 miles, and his dis- 
tance from tho sun 494,499,108 m.ilcs. 



I 



flvO Jtnt. 

. 234,507 
and, 

. 320,771) 
liiiii- 

. 709,813 
ami, 

. 1,217,457 
3vcn 

. 1,397,475 

54-7 • 
0105 
four 

. 2.30401 
tiou- 

. 000903 

. 4,370,000 

270 

0-50 

43 

. 000275 



W32C) 
10007 
M)405 
0075 

oooon 

040007 

^42 feet in a 
e same time, 
i in diameter j 
the earth. 
! 3,108 miles, 
I. 
and her dis- 

1 miles : it in 
md the latter 

, and her dis^ 

I his distance 

, and his dis- 



NOTATION AND NUAIKRATION. 



31 



) 



21. The diameter of Saturn is 78,730 miles, and hia die- 
taiico from the sun 907,089,032 miles. 

22. 'J 'ho length of a pendidmu which would vibrato 
Hcconds at the ci^uator, is 39011,084 inches; in the latitude 
of 4o degrees, it is 39116,820 inches; and in the latitude of 
90 degrees, 39-221,95G inches. 

23. It has been calculated that the distance from the 
earth to the nearest fixed star is 40,000 times the diameter 
of the earth's orbit, or annual path in the heavens ; that is, 
about 7,000,000,000,000 miles Now suppose a camion 
ball to fly from the earth to this star, with a uniform velocity 
equal to that with which it first leaves the mouth of the 
gun~-say 2,500 feet in a second— it would take nearly 
1,000 years to reach its destination. 

24 A p.iece of gold equal in bulk to an ounce of water, 
would weigh 19258 ounces; a piece of iron of exactly the 
sanie size, 7788 ounces; of copper, 8788 ounces; of lead. 
11'352 ounces; and of silver, 10474 ounces. 

NoTK.— The examples in notation may be made to answer 
for numeration ; and the reverse. 

QUESTIONS IN NOTATION AND NUMERATION. 

[The references at the end of the questions show in what 
paragraphs of the preceding section the respective answers 
are principally to be found.] 

1. What is notation } [1]. 

2. What is numeration .? [1]. 

3. How are we able to express an infinite iriety of 
numbers by a few names and characters ? [31 . 

4. How may we suppose ideas of numbers to have 
been origmally acquired > [4, &c.]. 

^5. What is meant by the common ratio of a system 
of numbers .> [12] . j ^^ 

6 Is any particular number better adapted than 
another for the common ratio .? [12]. ■ 

ratio .?^[11]^^''' '^'*'''^' ""^ numbers without a common 

8. What is meant by quinary, decimal, duodecimal, 
vigesimal, and sexagesimal systems ? [13]. 

9. Explain the Arabic system of notation ? ri5"I 

10. What are digits .? [161 ■^' 

1 1 TT *l 1 

. 1 . -^ow arc tliey maac to express all numberg ? [17] . 



I 



■^^ ti 



S9 



NOTATION AND NUMERATION. 



■*or of units of a lower order precisely the same thing ? 

14. Have the characters wo use, always and every 
Inhere been cmp oyed to express numbers ? [211 ^ 

rh.!i-ff ♦ ''i^' ^r^"^^^ P^>""*' «^»d the posit on of 

figiJesfe]."' ''^ '^ °^'"« affect- significant 
F^n'T,^''?^ !? the difference between the English and 

aolLIns^l '^^ '*^"^° '^^^^^'^ -*^g- -^ 

«.i^; T^'''* \' "'"''''* H *^^ ascending and descending 
X.^ [36] ' '" "^ "'" the/ related to each 

n„fJ' f '''V*^^* i". expressing the same quantity, we 
mist place the decimal point differently, according to 
the unit of comparison we adopt ? [38] 

22. What effect is produced on a digit, or a quantity 
by removing it a number of placs to the right, or left 
or similarly removing the decimal point ? [34 iid 39] 




^f. 



le and relativo 



33 



luivalent num- 
le samo thiujr ? 

ays and every 
' [211. 

:on? [22, &c.]. 
lie position of 
ICG to it ? [26 

3ct significant 

e English and 

)eriod8? [29 J. 

integers and 

nd descending 
slated to each 

quantity, we 
according to 

or a quantity 
right, or left, 
[34 and 39] 



SECTION 11. 

THE SIMPLE RULES. 

SlftlPLE AUDlTiON. 

oc,'; tLv""t'"ir "•'""«"'' J'y ""7 arithmotioal pro- 
ocas, they are cither morcasod or (iimiDislii.r) • if \„ 

f?™toe-££SS:?- 

but vo may have m mmm of tl,™ , ""^ """^' 

called "Multiplieator''\?l Zv'' ™ ^l'""'/'' '" 
.««, but Jheir'n„„.b?r'i. imlS^t' Ty™ I^^Jf Z 

quantities to be Uod-L^lnlL'T' "" 1";'"'"='' "^ *« 
til" kind l,„t ..„r.i ' '""'''Pl'ca'ion restricts us as to 

.•eally comprehended under trtl' ™^^SS' "" 

£.|s-^rn^i:-t^-r=: 

^..o%fu^';^iftiret;f.i,;L7;r^^^ 

means, that G is to be S d'od "o 8 wf "^ ^ ^ "" ?' 
prcfacd, the positive is undtstood '"" "" "^^ ^ 

to 16. ' ^^'^ ^'^^ ''^^" "^ 9 i^^^ 7 is equal 



II, ^if- 



34 



ADDITION. 



Quantities connected by the sign of addition, or that of 
equality, may be read in any order ; thus if 7 + 9=16, it 
is true, also, that 9 + 7=16, and that 16=7+9, or 9+7. 

5. Sometimes a single horizontal line, called a viii- 
mlum, from the Latin word signifying a bond or tie, 
is placed over several numbers ; and shows that all the 
quantities under it are to be considered, and treated as 

\)\xt one ; thus in 4+7=11, 4 + 7 is supposed to form 
but a single term. However, a vinculum is of little 
consequence in addition, since putting it over, or remov- 
ing it from an additive quantity — that is, one which has 
the sign of addition prefixed, or understood — does not 
in any way alter its value. Sometimes a parenthesis ( ) is 

used in place of the vinculum; thus 5+6 and (5+6) 
mean the same thing. 

6. The pupil should be made perfectly/ familiar with 
these symbols, and others which we shall introduce as 
we proceed ; or, so far from being, as they ought, a 
great advantage, they will serve only to embarrass him. 
There can be no doubt that the expression of quantities 
by characters, and not by words written in full, tends 
to brevity and clearness ; the same is equally true of the 
processes which are to be performed — the more con- 
cisely they are indicated the better. 

7. Arithmetical rules are, naturally, divided into two 
parts ; the one relates to the setting down of the quan- 
tities, the other to the operations to be described. We 
shall generally distinguish these by a line. 

To add Numbers. 

RirtE. — T. Set down the addends under each other, 
so that digits of the same order may stand in the same 
vertical celumn — units, for instance, under units, tens 
under tons, &c. 

II. Draw a line to separate the addends from their 
mm. 



III. Add the units of the same denomination together, 
boci:!nninf» at the rijzlit linnd side. 

IV. 9 the sum of any column bo less than ten, set it 
down under that column ; but if it be greater, for every 



ADDITION. 



35 



s from their 



ten it contains, carry one to tlie next column, and Dut 
down only what remains after deducting the'tens-^tf 
'wthing remains, put down a cypher ' 

V. Set down the sum of the last column in full. 

8. Example.— Find the sum of 542-|-375-^984— 

375 } addends. 
984 J 

1901 sum. 
Tt j'c . ' '^' ^"*^ ^' which are "hundreds" in nnnthoiT 

JO £t^pr:s^-i,--/-|SS 

Vn5? ^'""Pf' ^'ly"'" ^^^™^^ notation, can easily find 

l7d"Krof " '-: ^" ' ^^^"" ^^-^^r 5 sini all 
tiic dio ts that express it, except one to the ri^vht hnnrl 

j^ide, ml indicate the number of '4ens'' it JonVt „s 
thus m 14 there are 1 ten, and 4 units • in li^ S J ' 
-d 2 units ; in 143, 14 tens, and's unit's, L'' ' *"'' 
The ten obtained from the sum of the units alon<r wUh « 

:;reds._„„d write down a cypher in theTns. pLe" rf S^^ 
The two hundreds to be "carried " idflorl +a o Q i k 

As there are no thousands in the next cnlumn ^h.i- • 
ti.o last clnm i^'full. ""' ™*' ™ '<'' •^"™ "'« »"■» «f 

«lS-ri^Un<SLSimr^;rlL-! 




!i;!i 



36 



ADDITION. 



that we may easily find those quantities which are to be added 
together ; and that the value of each digit may be more clear 
from its being of the same denomination as those which are 
under, and over it. 

Reason of II.— We use the separating line to prevent the 
sum from being mistaken for an addend. 

Reason of III.— We obtain a correct result only by adding 
units of the same denomination together [Sec. I. 40] :— hun- 
dreds, for instance, added to tens, would give neither hnndreda 
nor tens as their sum. 

We begin at the right hand side to avoid the necessity of 
more than one addition; for, beginning at the left, the process 
would be as follows — 

542 
375 
984 

1,700 

190 

11 



1,000 
800 
100 

1 

1,901 



The first column to the left produces, by addition, 17 hun- 
dred or 1 thousand and 7 hundred ; the next column 19 tens, 
or 1 hundred and 9 tens ; and the next 11 units, or 1 ten and 
1 unit. But these quantities are still to be added :— beginning 
again, therefore, at the left hand side, we obtain 1000, 800 100° 
and 1, as the respective sums. These being added, give 1,901 
as the total sum. Beginning at the right hand rendered tho 
successive additions unnecessary. 

• ^\f^T'^^ OF IV.— Our object is to obtain the sum, expressed 
m the highest orders, since these, only, enable us to represent 
any quantity with the lowest numbers ; we therefore consider 
ten of one denomination as a unit of the next, and add it to 
those of the next which we already have 

After taking the " tens » from the sums of the different 
columns, we must set down the remainders, since they are 
parts oiihQ entire sum; and they are to be put under the 
CO umns tliat i.roduced them, since they have not ceased to 
Dclong to the denominations in these columns 

Reason or V.-It follows, that the sum of the last column 
ot„ \ ff * ^""y""^ '!" ^''^^ ' ^'^^' ('" *he above example, for in- 
it contains ""'' '" "°*^^"g *o be added to the tens (of hundreds) 

10. Proof of^Addition.—Oxxi off the upper addend, 
by a separating line ; and add tlie sum of tho (|uantitie» 



ADDITIOIf 



87 



re to be added 
be more clear 
ose which are 

;o prevent the 

mly by adding 
I. 40] :-~hun- 
ther hundreds 

e necessity of 
ft, the process 



tion, 17 hun- 
iumn 19 tens, 
, or 1 ten and 
I : — beginning 
000, 800, 100, 
id, give 1,901 
rendered the 

im, expressed 

to represent 

fore consider 

und add it to 

the diflferent 
Qce they are 
lit under the 
lot ceased to 

3 last column 
™ple, for in- 
9f liundreda) 

lor addend, 
(|uantitie» 






under, to what is above this line. If all the additions 
have been correctly performed, the latter sum will be 
equal to the result obtained by the rule : thus— 
5,673 

4,632 
8,697 
2,543 

21,545 sum of all the addends. 

15,872 sum of all the addends, but one. 
5,673 upper addend. 

21,545 same as sum to be proved. 
This mode of proof depends on «ie fact that the whole in 
equal to the sum of its parts, in whatever order they are 
Sf i' • Ji'*i' ^'""^l^. ^° *^® objection, that any error com- 
nutted in the first addition, is not unlikely to be repeated in 
the second, and the two errors would then conceal each other 

To prove addition, therefore, it is better to go through 
the process again, beginning at the top, and proceedinc 
downwards. From the princ^le on whicS the JtS of 
proof IS founded, the result of both additions-the direct 
and reversed— ought to be the same. 

It should be remembered that these, and other proofs of 

shfcf U ?, nnl '• ^^''"'^ '"'''^? "" ^'^^ d«g^«« «f pro'babiUty, 
since It is not in any case quite certain, that two errors cal- 
culated to conceal each other, have not been committed. 

ul'i^'^^^^^ Qwaw^2Vw!5 containing Decimals. —From. 
What has been said on the subject of notation (Sec. I. 
db) It appears that decimals, or quantities to the right 
hand side of the decimal point, are merely the continu- 
ation, doivnwards, of a series of numbers, aU of which 
to low the same laws ; and that the decimal point is 
mtendpd not to show that there is a difference in the 
nature of quantities at opposite sides of it, but to mark 

ni f- !i^v ""'^ 1 «TP""^°^" ^ Pl^^^d. Hence the 
mJo tor addition already given, r.pplies at whatever side 
a I, or any of the digits in the addends may be found 
it IS necea-ary to remember that the decimal point in 
the sum should stand precisely under the decunal points 
of the ^a.lends ; smce the digits of the sum must beffrom 
the very nature of tl^- — p°"=, rm a uT,oui.,num 

,, . ■' -''--^-'; '•' ^!-'- inuce.^3 [D , ot exactly the same 
value, rospectirely, as the digits of the addends under 

g2 




38 



ADDITION. 



whicli thoy are ; antl if set down as tlioy should bo, their 
denominations are ascertained, not only by their position 
with reference to their o i decimal point, but also by 
their position with reference to the digits of the addends 
above them. 

Example. 

263-785 
460-602 
637 -008 
626-3 



1887-695 

It is not necessary to fill up the columns, by adding 
cyphers to the last addend ; for it is sufficiently plain 
t^at we are not to notice any of its digits, until we come 
to the third column. 

12. It follows from the nature of notation [Sec. I. 
40], that however we may alter the decimal points of 
the addends— provided they are all in the same vertical 
column— the digits of the sum will continue unchanged ; 
mus in the followin<y : — 



4785 
8257 
6546 

14588 



478-5 
325-7 
654-6 

1458-8 



47-85 
32-57 
66-46 

145-88 



•4785 
•3257 
•6546 



1^4588 



•004785 
•003257 
•006546 

•014588 




I: 



EXERCISES. 

(Add the following numbers.) 



Addition, 


Multiplic 


(1) (2) 


(3) 


(4) (5) 


4 8 


3 


6 4 


6 4 


9 


6 4 


3 7 


7 


6 4 


6 6 


6 


6 4 


7 2 


5 


6« 4 


~~ — 


— 


— — 


_ — 


— 


— — 


(10) 


(11) 


(12) 


6763 


3707 


2867 


2341 


2465 


8246 


5279 


5678 


1239 



(6) 
9 

9 CO 
9 
9 
9 



(13) 
6978 
3767 
1236 



Involution, 

(8) 
4 



Ttt i 



(14) 
5767 
4579 
1236 



tO " 



(9) 
5 

6 
5 
5 

u 



(15) 
7647 
1239 
3789 



m 



ADOITJON, 



uld be, their 

ilieir position 

but also by 

the addends 



(16) 

6673 
123? 
2345 



(17) 
8767 
4567 
1^34 



(19) 

5147 

3745 
6789 



(20) 

34567 
47891 
41234 



39 



(21) 

73456 
4567? 
9123-J 



3, by adding 
iiently plain 
itil we come 

ion [Sec. I. 
al points of 
ame vertical 
unchanged ; 

•004785 
•003257 
•006546 



•0X4588 



(22) 


(23) 


(24) 


(25) 


(26) 


(27) 


76789 


34567 


78789 


34676 


73412 


36707 


46767 


89123 


01007 


78767 - 


70760 


46770 


12476 


45678 


34667 


45679 


47076 


36767 



(28) 


(29) 


(30) 


(31) 


(32) 


(88) 


45697 


76767 


23456 


46678 


23745 


87967 


87676 


45677 


78912 


91234 


67891 


32785 


36767 


76988 


34567 


66789 


23456 


64127 


(34) 


(35) 


(36) 


(37) 


(38) 


(39) 


30071 
45667 


45676 
37412 


37645 
67456 


47656 
12345 


76767 
12345 


45676 
34567 


12345 


37373 


123-15 


67891 


37676 


12345 


47676 


45674 


67891 


10707 


71267 


67891 



lution. 

8) 
^4 

4 

4 

4 



«5 " 



(9) 
5 

5 
5 
5 



- u 



(15) 
7647 
1239 

3789 



(40) 

71234 
12498 
91379 
92456 



(46) 

87376 
12677 
88991 
23478 



(41) 
19123 
67345 
67777 
88899 



(47) 

78967 
12345 
73707 
12(371 



(42) 

93456 
13767 
37124 
12156 



(48) 

34567 
12345 
7776G 
67345 



(48) 

45678 
34567 
12345 
99999 



(19) 

47676 
12345 
67671 
10070 



(44) 

45679 
34567 
12345 
76767 



(50) 

67678 
12345 
67912 

4G7()7 



(45) 

76766 
34567 
12345 
67891 



(51) 
67667 
34567 
23456 
76799 



40 



ADDITION. 



(52) 

76769 
12346 
76776 
466G6 



(53) 

57667 
19807 
34076 
13707 



(64) 
767346 

4767 of 
467007 
123456 



(55) 

478894 
767367 
412346 
671234 



(56) 

876767 
123764 
845678 
912346 



(67) 

676 

4689 

87 

84028 



(58) 

74564 

7674 

376 

6 



(69) 

5676 

1667 

63 

6767 



(60) 

76746 

71207 

100 

. 66 



(61) 

67674 

76670 

36 

77 



(62) 

42-37 

66-84 
27-93 
62-41 



(63) 

0-87 
6-273 
8-127 
?5-63 



(64) 

03-786 
20-766 
00-253 
10-004 



(66) 

86-772 
6034-82 

57-8563 
712-62 



(66) 

-00007 
-06236 
•0572 
•21 



(67) 
6471-8 
663-47 
21-60:^ 
0-00007 



(68) 

81-0235 
376-03 
4712-5 
6-53712 



(69) 

0-0007 
5000- 
427- 
37-12 



(70) 

8453-6 
-37 

8456-302 
•007 



(71) 

576-34 
4000-005 

213-5 
2763- 




m: 



72. £7654 + £50121 + £100 + £76767 4- £67^5 

=£135317. 

73. £10 + £7676 + £97674 + £676 + £9017 
=£115053. 

74. ^971 +£400+£97476+£30+£7000+£76734 
=£18261 1. 

T5. 10000 + 76567 + 10 + 76734 + 6763 + 67674-1 
=176842. -r"/D/-M 

76. 1 + 2 + 7676 + 100 + 9 + 7767 + 67=15622 
/7. 76 + 9970 -f 33 + 9977+100 f 67647 + 676760 
=764563. -rJiy>iuu 



ADDITION. 



41 



(67) 

676 

4589 

87 

84028 



(68) 

0-87 
6 -273 
8-127 
?5-63 



(67) 

ri-8 

33-47 

21-602 

0-00007 





(71) 

576-34 
000-006 
213-5 
753 • 





+ £675 

- £9017 

■£76734 

6767+1 

=15622. 
-676760 



m 



78. -75 + -6 + -756 + '7254 +'345 +'5 +'005 +-07 
• =3-7514. 

79. •4+74-47+37-007+75-05+747-077=:934-004. 

80. 5G-054-4-75 + -007+36-14+4-672=101-619. 

81. •76 + -0076 + 76 + -5 + 5 + -05.=82'3176. 

82. •5 + -05-i--005+5 + 50 + 500=:555'555. 

83. •367+56-7+762 + 97-6+471==1387-667. 

84. 1+-1 + 10 + '01 + 160+-001=17M11. 

85. 3-76 + 44-3+476-l-t-5-5=529-66. 

86. 36-77+4'42+M001 + -6=42-8901. 

87. A merchant owes to A. £1500 ; to B. £408 ; to 
0. £1310 ; to D. £50 ; and to E. £1900 ; what is the 
sum of all his debts ? A7i,s. £5168. 

88. A merchant has received the following sums : — . 
£200, £315, £317, £10, £172, £513 and £9 ; what ia 
the amount of all ? Ans. £1536. 

89. A merchant bought 7 casks of merchandize. No. 
1 weighed 310 tb ; No. 2, 420 ft ; No. 3, 338 ft ; No. 
4, 335 ft ; No. 5, 400 ft ; No. 6, 412 ft ; and No. 7 
429 ft : what is the weight of the entire ? 

Ans. 2644 lb. 

90. What IS the total weight of 9 casks of goods : 

Nos. 1, 2, and 3, weighed each 350 ft ; Nos. 4 and 5, 
each 331 ft ; No. 6, 310 ft ; Nos. 7, 8, and 9, each 
342 ft .? Am. 3048 ft. 

91. A merchant paid the following sums : — ^£5000, 
£2040, £1320, £1100, and £9070; how much was 
the amount of all the payments ? Ans. £18530. 

92. A linen draper sold 10 pieces of cloth, the first 
contained 34 yards ; the second, third, fourth, and fifth, 
each 36 yards ; the sixth, seventh, and eighth, each 33 
yards ; and the ninth and tenth each 35 yards ; how 
many yards were there in all .? Ans. 347. 

93. A cashier received six bags of money, the first 
hold £1034 ; the second, £1025 ; the third, £2008 ; tho 
fourth, £7013 ; the fifth, £5075 ; and the sixth, £89 ; 
how much was the whole sum .? Aiis. £16244. 

94. A vintner buys 6 pipes of brandy, containin<r as 
follows :— 126, 118, 125, 121, 127, and 119 galbns ; 
how many gallons in the whole ? A/as. 736 gals. 

95. What is the total weight of 7 casks, No. 1, con- 



42 



ADDITION. 



tainiug, om ib ; No. 2, 725 lb j No. 3, 830 ib • No 4 
VaBib; No. 6, 6«7 1b, No. B, 609 ib; and No.' 7,' 

Js como to r "'«'^1"'"<'"° «o^t ^39, what will 20 

S auTetlT' "''fy-«?^™i five thousand Lbm; 
t,™, ^^-^ ™' 'w thousand seven hundred anl 

tvm>ty.one ; hfty-s,. thousand seven hundred and seven^- 

inn \,M n ■!,■ ^''"- 206729644. 

fo„. .•■n- ""™.'"''I'««s and seventy-one thousand ^ 
four „,dhons and cghty-six thousand ; two mil ionTMd 

tweive" „£L a ' "'LXX^eno^u^lir, '™j 
-venty-two thou.,and, „i„e\„nd e'd a"d twen tlr™^ 

s:^,^t^^l3^£d5:s^eSS^ 

four hundred and ninety-one thou.,and. ^^J. 3 8700o' 
102 Add together one hundred and sixtv-sevon tl,m, 

dred 'a d'ii' V 'f T^-r™ 'housatdTZ tt 

fhi:;^; t ntruV, i'^^ntn^fdirrs 

■tno ,\,n ,1 ^ ,, -^^^^'. 3665000. 

iu,j. Add three tcn-tIionsan<ltlis • fo.tv fn,,,. r 
tenth, ; live hundredths ; six .hou.a.Uths, ti'ltlenltl "i;! 



ADDI'IION. 



43 



«andths ; four thousand aud forty K)no ; twcuty-two, one 
tenth ; one ten-thousandth. ' Ans. 4107*6r)72. 

104. Add one thousand ; one ten-thousandth ; five hun- 
dredths ; fourteen hundred and forty ; two tenths, three 
ten-thousandths ; five, four tenths, four tliousandths. 

Ans. 2445-6544. 

105. The circulation of promissory notes for the four 
weeks ending February 3, 1844, was as follows : — Bank 
of England, about iE21, 228,000 ; private banks of Eng- 
land and Wales, £4,980,000 ; Joint Stock Banks of 
lOngland and Wales, ii;3,446,000 ; all the banks of Scot- 
land, £2,791,000 ; Bank of Ireland, £3,581,000 ; all the 
other banks of Ireland, £2,429,000 : what was the total 
circulation ? Ans. £38,455,000. 

106. Chronologers have stated that the creation of 
the World occurred 4004 years before Christ ; the deluge, 
2348 ; the call of Abraham, 1921 ; the departure of the 
Israelites, from Egypt, 1491 ; the foundation of Solomon's 
temple, 1012 ; the end of the captivity, 536. This being 
the year 1844, how long is it since each of these events ? 
Ans. From the creation, 5848 years ; from the deluge, 
4192; from the call of Abraham, 3765; from the de- 
parture of the Israelites, 3335 ; from tlic foundation of the 
temple, 2856 ; and from the end of the captivity, 2380 

107. The deluge, according to this calculation, occur- 
red ] 656 years after the creation ; the call of Abraham 
427 after the deluge ; the departure of the Israelites, 
430 after the call of Abraham ; the foundation of the 
temple, 479 after the departure of the Israelites ; and 
the end of the captivity, 476 after the foundation of the 
temple. How many years from the first to the last ? 

Ans. 3468 years. 

108. Adam lived 930 years ; Seth, 912 ; Enos, 905 ; 
Cainan, 910 ; Mahalaleel, 895 ; Jared, 962 ; Enoch, 365 ; 
Methuselah, 969 ; Lamech, 777 ; Noah, 950 ; Shem, 600 ; 
Arphaxad, 438 ; Salah, 433 ; Hebor, 464 ; Peleg, 239 ; 
Eeu, 239 ; Serug, 230 ; Nahor, 148 ; Terah, 205 ; Abra- 
ham, 175 ; Isaac, 180 ; Jacob, l47. What is the sum of 
all their ages ? . Ans. 12073 years 

13. The pupil should not be allowed to leave addition, 



u 



ADDITIOIf. 



until ho can with groat rapidity, continually ad.l any of 

without hositation or furtL mention of the' numbtV 
J or instance he Bhould not bo allowed to proceeHhus : 

8 nr« ia"'' ^f i ^^ ""^ ' ^^-^ 21 ' ^'- J "«r even 9 a, ci 
b are 16 ; and a are 21 ; &o. Ho shoJld be able, uTu- 
raately, to add the following— ' 

6638 
4768 
9342 

1Q786 

in this manner :--2, 8 ... 16 (the sum of the column • 
of which 1 IS to be carried, and 6 to be set down) " s! 
10... 13; 4,11 ... 17; 10,14... 19. •' ' 

QUESTIONS TO BE ANSWERED BY THE PUPIL. 

reduce^d'?^[7]"''''^'"^'''"'^"" those of arithmetic be 

.2. What is addition .? [3J. 
tion ? ^3^* ""^^ *^' ''''°''' '^ *^' quantities used in addi- 
t' WkI ^^\*^' -^S"? °^ ^^'^^*^'^°' ^^^ equality ? [41 

ndditir;Utit rs"^^^ ^ ^^^ ^'^' - ^^^ ^^-^^ - 

^* Su^* ^^ *^*® ^'^^^ ^°^ addition ? [7] 

7. What are the reasons for its different parts > [91 

8. J^es this rule apply, at whatever side of the deci- 

fo^dT[llj"' "' '"^ '' *^^ ^"^"^^^^^« '' ^^ -dded a"e 

9. How is addition proved ? [10]. ^ 

10. What is the reason of this proof.? [10]. 



8UUTRACTI0N. 



46 



SIMPLE SUBTllACTION. 

14. Simpk siibtraction is confined to abslract numbers, 
and apphuate which consist of but one denomination 

^subtraction enablcH us to take one number called' tho 
subtrahend, from another called the minuend. If anv- 
tlnng s loft, it is called the excels ; in commercial con- 
cerns, It IS termed the remainder ; and in the mathema- 
tical sciences, the difertmce. 

15. Subtraction is indicated by —, called the minus, 
or negative sign Thus 5-4=1, read five minus four 
equal to one, mdicates that if 4 is substracted from 6. 
unity is left. » 

Quantities connected by the negative sign cannot be 
taken, indifferently, m any order ; because, for example, 
6-4 is not the same as 4—5. In the former case the 
positive quantity is the greater, and 1 (which means 
+ L4J) IS left; m the latter, the negative quantity 
18 tlie greater and -1, or one to be subtiactld, still 
remams.^ To illustrate yet further the use and nature 
ot the signs, let us suppose that we hmx five pounds 
and owe four;— the five pounds we hate will be repre- 
sented by5 and our debt by -4 ; taking the 4 f?om 
the o, we shall have 1 pound ( + 1) remaining. Next 
let us suppose that we have only four pounds and owe 
five ; If we take the 5 from the 4-that is, if we pay 
a. fkr as we can-a debt of one pound, represented by 
--1, ^111 still remain ;— consequently 5—4=1 ; but 

nr.^^' V'"'"''^"'? placed over a subtractive quantity, 
or one having the negative sign prefixed, aiteTs its 
value, unless we change_all the^igSs but the first •- 
thus 5-3+2, and 5—3+2, are not the same thing- 

^If -^t^^'* ^ but 5-3+2 (3+2 being considered 
nowas but one quantity) =0 ; for 3+2=5 ;-therefor« 
j-3+2 IS the same as 5-5, whifA leaves nothing ; or. 

m herwords, it is equal to 0. If, however, we cli'an.e 

all the ai'T'"° ^-^^r^^*- ai.~ x?-_i .i i _ . ' ^""^,0 

i- -uc sj^,.,.j vAvcpt ixi-c m-Ki, ine vaiuo of the quantity is 



l^M 





4R 



••UHTRACTIOPI. 



uot aUci-o(l by flit) viiKMilmn ;— thus 5-3-^2=4; and 
f*~-3 — 2, also, Is c(iual to 4. 

Again, 27-44-7— 3=27. 
27- 

But 



■4+7-3=19. 

27—4—74-3 (chaiifyinip all tho ilgng of the ) OT 
' ori((iuuI quantitloi, but tha flnt) { ■*■*• • 



The following examplo will show how the vinculum 
attecta numbers, according us wo mako it include an 
additive or a subtractiv<3 quantity : 

48-f- 7-3-8-f-7-2 =49. 

48-1-7 — 3 — 8 JL7_ 2=49 • what ia under the vinculum beinjf 

' additive, it is not necessary to 
, change any signs. 

48-f-7— i5-l-8 — 7-1-2=49 • ^' " ""^ necessary to change aU the 
d^_L7 q sr~"Tio ,n ',«.'«"» «»'^<'"tlie vinculum. 
^'^"r'~'J— "— 7-f-2 =49; it is necessary in this case, alto, 

48-1-7-3-8-1-7=2=49; u ^^^^J^^j^^,, ease. 

In the above, we have sometimes put an additive, and 
sometimes a subtractive quantity, under the vinculum ; 
in the former case, wo wore obliged to change the signs 
ot all the terms connected by the vinculum, except the 
hrst— that IS, to change all the signs under the vin- 
culum ; m the latter, to preserve the original value of 
the quantity, it was not necessary to change any sign. 

To Subtract Numhers. 

„nrl!" ^""'^TI: ^^^'' ;^'' ^^«^*« «f *^« subtrahend 
under those of the same denomination in the minuend— 
units under units, tens under tens, &c. 

II. Put a line under the subtrahend, to separate it 
iroin the remainder. 



III. Subtract each digit of the subtrahend from tho 
one over it m the minuend, beginning at the right hand 

IV. If any order of the minuend be smaller than tho 
quantity to be subtracted from it, increase it by ten : and 
cither consider the next order of the minuend as lessoned 
by unity, or the next order of the subtrahend as in- 
creased by it. 

V. After subtracting any denomination of the sub- 



suiirriACTiOiV. 



47 



h2=:4; and 



lie flnt) ! =27. 

he vinculum 
include an 



vinculum hein^ 
ot aeoessary to 
1. 

to change all the 
inciilum. 
thia case, also, 
ns. 
in this case. 

(Iditivc, and 
I vinculum ; 
;e the signs 
except the 
Icr the vin- 
al value of 
any sign. 



subtrahend 
minuend — 

separate it 



1 from the 
right hand 

5r than the 
J ten ; and 
IS lessoned 
end as in- 

f the sub- 



trahend from the correspoudiug pjut of the nunuond 
H',;fc (l.)wii wh.-it i,s loft, if liny thing, in the phiee which' 
b<.'li)ii(;s to thi! Kaiiie donroiiiri.-itioti (»f the " rcnuiindfr." 
Vr. ]Jat if th(!rc \a no\\nn<^ Idl, put down a cyphoi-- 
provided any digit of the " rcMuuiudor" will be niore dis- 
tant from the deciiuul point, and ut tlio same side of it. 
18.. K.\A.MrM.: l.~8ubtr;ict 427 from 71)2. 
n)2 minuend. 
427 «ul;tnilu!iul. 

u(J5 remainder, ditterenco, or ox<'(!,s«<. 
Wo cannot take 7 units iunxx 2 unitfl; hut "bomnvin-." nfl 
It IS calind, one ot the !) tons in the )ninti(Mi(l, ,uid consi,],.,-. 
«ng It as /6'u unit.s. wo add it to the 2 units, and tbnn liavo 
1- units; taking / Iroiii 12 unit.s, 5 arc left:- wo put o in 
the units pace o/ the "n^maindor." Wo may considor tlio 
.' tens ot tho niinuend (one luiving been taken away, or 
borrowed) as 8 tons; or, which is the same thin-. 'may 
suppose the I tons to remain as they wore, b-.t tho''2 tcn"^ 
ot tho subtrahend to hi.ve beoomo [\; tlicn, 2 tens from « 
t.-ns or o tens from U ten.s, and tens are lell :-wo i.ut i\ 
in the tens' place of tiie "remainder.- 4 hundreds, of tho 
Hubtrnbend, taken from the 7 Junidreds of the ininurud,' 

ExAMPLK 2.— Take 5G4 from 7G8. 

7G8 
504 

2U4 
When G tons are taken from G ten.s, notliing is Irft : w,. 
therefore put a cypher in the tens' place of the ^-emainder" 
KxAMi=j.E 3.— Tako 537 from 5U4. 

594 
537 



s: 



When 5 hundreds are taken from 5 hun.livds notliln.. 



48 



SUBTRACTION. 



Of the subtrahend maybe near those of the mmuend from 
which they are to be taken ; wo are tlien sure that the JrL^ 
Sfriuff^Evfl-*'' subtrahend and nainueuS ^^ybe 
douMn. fo1>,n^^^ "^ arrangement, also, we remove any 
1„ ^ \ , *^^ denominations to which the diVits of the sub- 
trahend belong-their value? being rendered more cert^Lbv 

S;SZ Tf rf''''' '' the'cligits of th'rnS"' '^ 
Keason OF H.-The separating line, though convenient i- 
not of sueh importance as in addition [9] ; si^io th^" remain 
der » can hardly be mistaken for another quantity 
Keaso^t of Ill.-When the numbera are considerable 

powe"' ?tt"mii^^^^^^^^ be effected at once, from'ThflSd 
poweis ot tne mind; we therefore divide the ffiven ouantitipq 
into parts; and it is clear that the sum of the d'ffe?en?es of 

th^ sTmTrS"/ ''T' ^f, ^"l^i^ *^ *»- diffe're^r Ween 
to 500 4nl7n 9^TS =7*^"'"' ^^^-^27 is evidently equal 
rebble^"^?"^ w7?.+^~J\r "•''? ^^ «h«^» *« the child by 

be necesstrv tonU. ^'"^ ^* "ilP^^*. ^^^'^ ''^^' ^^cause it maj 
oe necessaiy to alter some of the d g ts of the minuend so a*^ 

to make it possible to subtract from them the corresnondin^ 

ones of the subtrahend; but, unless we beSn at the STt h3 

Bide, we cannot know what alteratioi-s may be iVquired ''''^ 

thin'tlfo'n '" '^-T^' ",^^. ^"g^* '' tlie iinuend be smaller 
than the corresponding digit of the subtrahend, we can proceed 

l?on fTl f *•'" ""T-. .*;"''*' ^« ^'-^y i^^rease .hat denmina- 
tion ot the minuend which is too small, by borrowing <,nrfrom 

or'tE v^^Ms t^obf'''^' ""'. 'r f t^e lower InlSaL^ 

p^|^:ii; i^ tr^^. ^i^^^^r^it^ s 



idrcds. 


tens. 


units. 


7 

4 


8 
2 


12 = 792, the minuend. 


/ = 427, the subtrahend 



^ 5 = 305, the diflference. 

an^'r,^wV' "I® "['^y «dd equal quantities to both minuend 
woulfw '"'^' ^'^^"^ ^^" ^°^ ^^^-'- the difference; tien we 



HuiKireds. tens. 
7 9 

4 2 + 1 



6 



units. 

2 4- 10 == 792 -f 10, the minuend -f 10. 
7 = 42 7 -f 10, tlie subtrahend -f 10. 

== 365 -|- 0, the same difference. 



ini«!fKf"'''^ ""a T'!''*'?"-? ^« ^0 not use the given minuend 
and subtrahend, but others which produce the ^ImTvZZl^T, 

Reason of V.— The remainders obtained bv subtraetin^ 
successivfilv. fl.o riiffn«„„4. .i„„_. •, ,. "'^ - "/ suDtracting, 

froai those which correspond in the minuend are the i^ari of 






,1 ij," 



SUBTRAC'IION. 



> minuend from 
that the corres- 
linueud may be 
WQ remove any 
gits of the &ub- 
nore certain, by 
s minuend. 
1 convenient, is 
ie the " I'cmain- 
tity. 

3 considerable, 
pom the limited 
;iven quantities 
e differences of 
jrence between 
Jvidontly equal 
;o the child by 

because it may 
minuend, so as 
s corresponding 

the right hand 
equired. 
nd be smaller 
we can proceed 
ihat denoraina- 
iwing one from 

denomination, 
to those of the 

we alter the 
, in the exam* 



uend. 
trahcnd. 

rence. 

both minuend 
Jnce ; then we 



id -f- 10. 
lend -f 10. 

ifforence. 

iven minuend 
no remainder, 

subtracting, 
ts Bubtrahend 
i the jparts of 



4Q 



the total remainder. They are to be set down under the .Ipnn 

20 Proof* of SvMradion.—Add to^-.her the re- 
mainder and subtrahend ; and the mm sliould be equal 
to the minuend. For, the remainder expresses by Lw 

£hf' *t 'f ' r""^"^'^«r to the subtrahend 

should make it equal to the minuend ; thus 

8754 minuend. 
6839 subtral. nd. ^ 

2915 difference. ) 
Sum of difference and subtrahend, 8754=.minuend. 

wh?t^^tf^'^ */f /^"^^^^*»1*^^ f'-*^"^ ^^^e minuend, and 
f hp I ' 7 -"^^1 ^' "'^"'^^ ^" ^^'' subtrahend. For 

8G84 minuend. Pun^r. . aroA ^- i 

rciQK 1 i 1 , irRooF : obrf4 minuend 

i98o subtrahend. J549 remafnder. 

649 remainder. New remainder, 7985=subtrahend 

8034 minuend. 

7985 subtrahend. 
T,.„, , , 049 remainder. 

Difference between remainder and minuend, 7986= subtrahend 

T^mV' Tho^lT''-*\'^'^'f'' *■". ^^'^^'^■f^^^^^ contain Deci^ 
s do "i^fl r • ^T^ ^''^'V ^PPli^'-^t'J^, at whatever 
k fold tv'TnP'"'^ '"^^ '' '"^"^'^^ ^^^« ^i^^^ts may 
VMT '7.;' follows, as in addition [11], from the 

Y'y n.Uuie of notation. It is necessary to put th.= 
decimal point of the remainder under the IZS'JnU 

v^lZ.lT! fl "; '""S ^^ ^^'""^ ^*"-^^t, have the samo 
^alue as the digits from which they have been derived. 




60 



Example. 



SUBTRACTION. 

-Subtract 427-85 from 503-04. 

663-04 

427-85 

135-19 



Since the digit to the right of the decimal point in the 
onthT' ?■' ''"'^''''''' '"^"* '' ^'^'' ''^''' *^»« subtraction o i.o 

t e inhtl n' T^""^ P?mt indicates vvimt remuins^fter 
the subtraction of tlie units, it expresses so many units:- 
all this IS shown by the position of the decimal point. ' 

An^^'J\ ^y^'''^'' ^'''™ *^^ principles of notation [Sec. I. 
40J, that however we may alter the decimal points of 
the mmuend and subtrahend, as long as the/stand in 
the same vertical column, the didts of the difference 
are not changed ; thus, in the following examples, the 
fiame digits are found in all the remainders •— 



4362 
8547 

815 



436 
354' 



81-6 



43-62 
35-47 

8-15 



•4362 
•3547 



•0815 



•000 J 3 62 
•0003547 



•0000815 



EXERCISES IN SUBTRACTION. 



. From 
Take 


(1) 

1969 
1408 


(2) 
7432 
6711 


(3) 

9076 
4567 


(4) 
8146 
4377 


(5) 

3176 

2907 


(fi) 
76877 
45761 


Froi.i 
Take 


(7) 

86167 
61376 


(8) 

67777 
46699 


(9) 
71234 
43412 


(10) 
900076 
899934 


(11) 
376704 
297610 


(12) 

745674 
376789 


From 
Take 


(13) 

67001 
35690 


(14) 
9733376 
4124767 


(15) 

567i)74 
476476 


(16) 
473(i76 
S21799 


(17) 
6310756 
3767016 


(18) 
376576 
240940 



m 



SUBTRACTION. 



il point In the 
)ti-acti(m of tho 
ice the digit to 
i remsiiiis after 
many units; — 
al point. 

:ation [Sec. 1. 
nal points of 
they stand io 
;he difference 
3xamples, tlie 
3 : — 

•0001362 
•000;J547 



•0000815 



(19. 
From 345070 
Take ]799 



(20) 
234100 
9U0 



(21) 

4367676 
25G560 



(22) 

845073 

I2479'J 



(23) 

70101076 
37091734 



61 

(24) 

67300000 
31237777 



From 
Take 



(25) (26) (27) (28) r29) no^ 

S?? JS f= ??S = ~ 



47134777 1123640 7476909 



(31) 

From 7045076 
Take 3077097 



(36) 

From 11000000 
Take 9919919 



(82) 

87670070 
26716645 



(37) 
3000001 
2199077 



(33) 

70000000 
9999999 



(38) 

8Q00800 
(J77776 



(34) (35) 

70040500 60070007 
56767767 41234016 



(39) 

8000000 
62358 



(40) 

404006b 
220202 



(5) 


(6) 


n-Q 


76377 


i907 


45761 



I) 

704 
610 



) 



(12) 

745674 
376789 



(18) 



'50 370570 
)16 240940 



From 
Take 



(41) 

85-73 
42-16 



(46) 

From 0-00003 
Take 0-00048 



(42) 

805 -4 
73-2 



(47) 
874-32 
6-63705 



(43) 

694-763 
85-600 



(48) 

67-004 
2-3 



(44) 

47-030 
0-078 



(49) 

47632- 

0-845003 



(45) 

52-137 
20-005 



(50) 

400-327 
0-0006 




745676—507456=178220 

500789—75074=501115.* 

941000-5007=935993. 

9/001—50077=40024 

70734-977=75757. 

56400-100=50300. 

700000—99=099901. 

5700—500=5200. 

9777—89=9088. 

70000-1=75099. 

90017-3=90014. 



02. 97777-4=97773. 

03. 00000-1=59999. 

64. 75477—76=76401. 

65. 7-97- 1-05=6-92. 
00. 1-75— .074=1-676. 

07. 97-07—4-709=92-301. 

08. /• 05— 4-776=2-274. 

09. 10-701—9-001=1-76. 

70. 12-10009-7-121=4-07909 
n. 170-1 — •007^-..176-093.' 



72. 15-00 



-< -803=7-197. 



53 



SUBTRACTION. 




73. What number, .iddcd to 9709, will make it 10901 

oaJ: ^J^^*"*^^' ^"^1^t 20 pipes of hrandj, containing 
2459 gallons, and sold 14 pipes, containing 1680 gal- 
Ions ; how many pipes and gallons liad he remaining ? 

Ans. 6 pipes and 779 gallons. 

75. A merchant bought 664 hides, weighin^r 16800 

Jb, and sold of them 260 hides, weighing 78091b ; how 

many hides had he unsold, and what was their wei^rht > 



76. 



Am. 304 hides, weighing 8991 lb 
A gentleman who had 1756 acres of land, gives 
2o0 acres to his eldest, and 230 to his second son ; how 
many acres did he retain in his possession > Ans. 1276 

77. A merchant owes to A. i^SOO ; to B. £90 • to D 
|7o0; toD. ^600. To moot tl.co' debts t has but 
d;.971 ; how much is he deficient ? Ans jei269 

78 Paris is about 225 English miles distant from 
London; Eonie, 950; Madrid, 860; Vienna, 820- 
Copenhagen, 610; Geneva, 460 ; Moscoav, 1660 : Gib- 
ral^nr, 1160; and Constantinople, 1600. How much 
more distant IS Constantinople than Paris; Rome than 
Madrid ; and Vienna than Copenhagen. And how much 
less distmit IS Geneva than Moscow; and Paris than 
Madrid } Am. Constantinople is 1375 miles moro dis- 
taut than Pans; Rome, 90 more than Madrid; and 
V lenna, 210 more than Copenhagen. Geneva is 1200 
miles less distant than Moscow; and Paris, 635 less 
tnan iuadnd. 

79. How much was the Jewish greater than the 
J^-.ngIish mile ; allowing the former to have been 1-3817 



miles Endish > 



80. 
mile ; 
mile 5 

81. 



Am. 0-3817. 



How much IS the English gi-eater than the Roman 
allowing the latter to have been 0-915719 of a 

T#u'^-' .1 , . ^^^- 0-084281, 

W hai IS the value of 6 - 3 + 1 5 - 4 .? Am. 1 4 

Afis S'? 

Of 47-6-2+1-244-16--34 } Am. 52 94 

84. What is the differencc betwccn 15+13—6—81 + 

Am. 38. 



S2. Of 43 + 7-3^^? 

83. 

84. __ ^ 

02, and 15+13—6=11 + 62 } 



23. 



Before leaving this rule, the pupU should "be able 



;j 



i 



M 



"* 



MULTIPilCATlON. 



53 



:c it 10901 
Alls. 1192, 
5 containing 
; 1680 gal- 
:nainiug ? 
'79 gallons, 
iing 16800 
09 It) ; how 
eir weight ? 
ig 8991 lb. 
land, gives 
I son ; how 
Ans. 1276. 
£90 ; to C. 
be has but 
ns. jei269. 
stant from 
ma, 820 ; 
660 ; Gib- 
low much 
lome than 
how much 
Paris than 
moro dis- 
Jrid ; and 
a is 1200 
, 635 less 

than the 
an 1-3817 
. 0-3817. 
iie Roman 
)719 of a 
)-084281 
Ans. 14 
Ans. 33 
IS. 52 94 
•6—81 + 
Ans, 38. 

I be able 



to take any of the nine digits continually from a given 
number, without stopping or hesitating. Thus, sub- 
tracting 7 from 94, he should say, 94, 87, 80, &c. ; and 
should proceed, for instance, with the following exampla 

5376 

4298 



1078 



m 



this 



manner :~8, 16.. .8 (the difference, to be set 
down); 10, 17...7; 3,3...0; 4, 5...1. 

QUESTIONS TO BE ANSWERED EV THE PUPIL. 

1. What is subtraction .? [14]. 

2. What are the names of 1*e terms used in subtrac- 
tion ? [14]. 

3. What IS the sign of subtraction ? [15]. 

4. IIow is the vinculum used, with a subtractive 
quantity? [16]. 

5. What is the rule for subtraction .? [17]. 

6. What are the reasons of its different parts.? [19]. 

7. Docs it apply, when there are decimals ? [211 



tion 



8. How is subtraction proved, and wliy .? [2oT. 

9. Exemplify a brief mode of performiu'^ su 
" ' [23] ^ ° 



subtrac- 



SIMPLE MULTirLIGATION. 
24. Simple multiplication is confined to abstract 
numbers, and apphcate which coataiu but one denomi- 
nation. 

,».^?r ^^'?^'"" '"f '^'' "' f" ''^^^ ^ ^"^'^ity. ^'^^"cd the 
^ ¥^anrl, a number of times indicated by the ,nM- 

ittl U ^''f'^'^''' .'% t'^^t h wlucli we multiply : 
the x^sult m he multiplication is called the jJdt. 

addo d,'' ,n multipheatiou, is termed the <' multipli ' 

JZh^ !^' '^^ '' designatoa tho " product.'' The 
Uuautu.es which, whou muitipljcd log.^l|cr, give tho 

' ' J.) 



m 



Ml 




64 



MULTIPLICATION. 



poduot, nro cnlle.l also factors, nnd, when they ,re 

•S'^rtf^^'"* J^^^'^^ ;"/^-^ ^- — than' two 
actors m that case, the multiplicand, niultinlier or 

b and 7, be the factors, either 6 times 6 may be con 

^^V^^;/""Itiphcand, and 6 times 7 as the mulLLr 
2o. Quantities not formed by the continued addS^on 
of any number, but unity-that is, which are not h^ 
product^; of any two numbers, unle'ss unityTs taken as 
one of them-are called privii numbers : Yll ot ts are 
termed co^nposiie. Thus 3 and 5 are p me but 9 
and 14 are composite numbers; because oniyXJ 
midtipbed },y...,,will p«pduce "'three," aU oKi 
mu tip led by one, will produce " five, "-but //S 
multi^hed by three will produce " nine," ind seTeSi mil 
tiphed by hvo will produce " fourteen " 

fl!.r. \ ■*PT^'^ ^^ ^" e«^%-er-or, in other words 
matder i^ f'^^r^f ^^"^^ ^^ ^^^^^^^ leaving a rei 

contabed in I .1 "" '\- ^""'^^'^ "^ ^^^ ^'^^^'^ it i« 
Pd from if a^t^b^^f r^^pi-eS b'yNt" 

measure ot 14, because, taking t as often as r^n^^ihC. 
from 14 4 will .till bo lefti-this, I.3_3=,0, /o-S 



-5=0, but 14—5=9, and 9—5=4. 



Measure, 



', ,:. , ' "," ^^ — '-'^ry, una y — 5=4. 
submultiple, and aliquot part, are synonymous. 

is V numllrTrr V'''' '^ *^" ^^ ^^°r« q^^^tities 
IS a number that will measure each of them • it is a 

measure comnon to them. Numbers which^ have no 

ocner all otliers are comj^^^z^e to each other. Thus 7 
and 5 are i;r^/;^e to each other, for unity alone will 

b::s:3':iiV ' ^^^^ ^^^- ^^^^^sr^ :^,^^ 

uecause 3 wUl measure either. It is evident that two 

ITi ^Z^'l ""f "= p™- "> -^''' *" , ttfs 

e°ceM „n^t. fr"' •""''''"'■'', •'"™"' "»• '^ ""•'-■. ''"<'- 



MUI,TIPLICATIO^^ 



65 



on they are 
I'e than two 
lultiplier, or 

Thus, if 5 
May be con- 
iiltiplier — oi 
nultiplier. 
led addition 
ire not the 
is taken as 
I others are 
ime, but 9 

only tkreey 
id onlyfivey 
-but, three 
I seven mul- 

tne number 
thor words, 
ving a ro- 
ot pari of 
3cause it is 
Q be sub- 
d by 3, an 
) is not a 
IS possible 

I, JO— 5r= 

Measure, 

• 

quantities 
1 : it is a 
have no 
'Me to each 
Thus 7 
tlone will 
ch other, 
that two 
r thus 3 
:o, and — 
will niea- 




■"fti 



Two numbers may bo oninpnsito to each other, and 
yet (yne of them may be a fvme number ; thus 5 and 25 
are both measured by 5, still the former is ytrim. 

Two numbers may be composite, and yet prime to 
emh other ; thus 9 and 14 are both composite numbers, 
yet they have no covwion measure but unity. 

28. The greatest common measure of two or more 
numbers, is the greatest number which is their common 
measure ; thus 30 and 60 are measured by 5, 10 15 
and 30 ; therefore each of these is their ccmimon mea- 
sure ; — but 30 is their greatest common measure. When 
a product is formed by factors which are integers, it is 
measured by each of them. 

29.^ One number i- the i^uUiple of another, if it 
contam the latter a number of times expressed by an 
integer. Thus 27 is a multiple of 9, because it con- 
tams It a number of times expressed by 3, an integer 
Any quantity is the multiple of its measure, and the 
measure of its multiple. 

^ 30. The com7non multiple of two or more quantities, 
IS a number that is the multiple of each, by an intcffer •-- 
thus 40 IS the common multiple of 8 and 5 ; since it is a 
multiple of 8 by 5, an integer, and of 5 by 8, an integer. 
^ LhY<^st common multiple of two or more quantities, 
IS the /m5^ number which is their common multiple--, 
thus 30 IS a common multiple of 3 and 5 ; but 15 is 
then- least copunon multiple ; for no number smaller 
tnan lo contains each of them exactly. 

31. The equivmltiples of two or more numbers, are 
then- products, when multiplied by the same number ;— 
tlius 27, 12 and IS, are equimultiples of 9, 4, and 6 • 
because, multiplying 9 by three, gives 27, multiplying 4 

^^4f 'tV^u-'r^-'.''^^ ^^^itiplying 6 by three, give^ IS. 

S2 Multiplication greatly abbreviates the process of 
addition ;— for example, to add 68965 to itself 7000 times 
})y audition," would be a work of great labour, and con- 
sume much time ; but by " multiplication," as we shall find 
presently It cn^i be done with case, in less than a minute. 

roi.i 7 i?v "•''^ .'"'"' inaccurate, to have stated 
L-^J that multiplication is a species of addition ; since we 
can know the product of t^vo quantities without havin^' 




06 



MULTIPLICATION. 



recourse to that rule, if tliey are found in the multipli- 
cation table ?..t it must not be forgotten that the mul- 
plioation table IS actually the result of additions, long 
since made ; without its assistance, to multiply so simplf 
a number as 4 by so smaU a one as five, we shodd be 
obliged to proceed as follows, 

4 
4 
4 
4 
4 

20 

performing the addition, as with any other addends 
The multiplicat on tabl^is due to Pythagoras, a" cele- 

bSortcS '''''^''''^ ^^^ ^- ^-^ ^^^ y-s 

34. We express multiplication by X ; thus 5x7— 

thatXT *^\^r^*f-d by 7 ire equal tol's, ^^ 

that the product of 5 aTid 7, or of 5 by 7, is equal to 35 

When a quantity under the vinculum 'is toTe muf^ 

plied-for, to multiply the whole, we must multiply 
eack of Its parts^:— tlms^^7+8=3=3X7+3xS-l 
3X3; and 4+5X8+3-6, means that each of the 
terns under thelaiier vinculum, is to be multiplied by 
each of those under the former ^ ^ 

maf be^ro^ltf ""'''''"' f ^^ ?^' ^^^n of multiplication 
may be lead in any order; thus 6X6=6X5 This 

Will be evident from the foUowing mustration, by which 
of it :-! ' ^ ' ^'^^^^^"S to the view we take 



«> 



8 

♦ ♦ • 

♦ ♦ • 

♦ • • 
» • » 



"^ ^ » • 

Quantities connected by the mgn of multipUcation, 



MULTIPLICATION. 



Jie multipli- 
bat the mul- 
tlitiona, long 
ly so simple 
e should be 



67 



Idends. 
>ras, a cele- 
590 years 

us 5x7= 
I to 35, or 
qual to 35. 
) be multi- 
be multi- 
t multiply 

r+3xs— 

ch of the 
Jtiplied by 

ttiplication 
<5. This 
. by which 
Bonsidered 
y we take 



V 



't 



plicati^ 



ion. 



pre multiplied if we multiply one of the factors ; thus 
GX7X3 multiplied by 4=6X7 multiplied by 3X4. 

36. To prepare him for multiplication, the pupil 
should be made, on seeing any two digits, to name their 
product, without mentionhig the digits tiiemselves. Thus, 
a largo number having been set down, he may begin 
mth the product of the first and second digits; and 
then proceed with that of the second and thh-d, &c! 
Taking 

587C349258G7 
for an example, he should say: — 40 (the product of 5 
and 8) ; 56 (the product of 8 and 7) ; 42 ; 18 ; &c.,as 
rapidly as he could read 5, 8, 7, &c. 

To Mibltijply Nmibers. 

37. When neither multiplicand, nor multiplier ex- 
ceeds 12 — 

Rule. — Find the product of the given numbers by 
the multiplication table, page 1. 

The pupil should be perfectly familiar with this table. 
^ Example.— What is the product of 5 and 7 ? The mul- 
tiplication table shows that 5x7=35, (5 times 7 are 35). 

38. This rule is applicable, whatever may be the 
relative values of the multiplicand and multiplier ; that 
is [Sec. I. 18 and 40], whatever may be the kind of 
units expressed— provided their ahsolwte values do not 
exceed 12. Thus, for instance, 1200X90, would come 
under it, as well as 12X9 ; also •0009X0-8, as well as 
9X8. We shall reserve what is to be said of the man- 
agement of cyphers, and decimals for the next rule ; it 
will be equally true, however, in all cases of multiplica- 
tion. 

39. When tlie multiplicand does, but the multiplier 
does not exceed 12 — 

Rule. — I. Place the multiplier under that denomi- 
nation of the multiplicand to which it belongs. 

II. Put a line under the multiplier, to seplirate it from 
the product. 

ni. Multiply each denomination of the multiplicand 
by tiie multiplier— bogiuniii;;^ ut the rit^'ht hand side. 



--+1 



08 



MULTIPLICATIOX. 




I\. If tlm prodiipt of tho multiplier and any digit 
of tho liiultiphcand is Ichs than ten, set it down under 
that (tigit ; but if it bo greater, for every ten it contains 
carry one to tho next produ.., and ])ut down only what 
remains, after d. o' u tir. the tens; if nothing remains, 
put down a cypher. ' 

,V. Set down the last product in full. 

40. Example. 1.— What ia the product of 897351x4? 
SOTI^Sl multiplicand. 
4 multiplier. 

3581)404 product. 
4 times one unit are 4 units; since 4 is less than ten, it 
gives nothing to be "carried," we, therefore, Bet it down n 
the units' place r f the product. 4 times 5 are twenty (tens)-: 
which are equal to 2 tens of tens, or hundreds to I o carried, 
and no units of tons to be set down in the tens' place of 
the product— in which, therefore, we put a cypher 4 
times 3 are 12 (hundreds), which, with the 2 hundids to bo 
carried from the tens, make 14 hundreds; these are equal 
to one thousand to bo carried, and 4 to be set down in the 
thousan<l8' pluee of the product. 4 times 7 are 28 (thou- 
sands), and 1 thousand to be carried, are 29 thousands ; or 
2 to be carried to tho next product, and 9 to be sot do\vn 
4 times 9 are 3b, and 2 are 38 ; or 3 to be carrriod, and 8 to 
be set down 4 times 8 are 32, and 3 to be carried are 35 ; 
which 13 to be set down, since there is nothing in the next 
denomination of the multiplicand. 

Example 2.— Multiply 80073 by 2. 

80073 
2 



16014G 
Twice 3 units are units ; G being less than ten, gives 
nothing to be carried, hence we put it down in the units' 
place of the quotient. Twice 7 tens are 14 tens; or 1 ' undrod 
to be carried, and 4 tens to be set down. As there are no 
hundreds in the iiadtiplicand, we can have none in the pro- 
duct, except that whicli is derivtsl from the multiplication 
ot the tens ; we accordingly put the 1, to be carried, in the 
hundreds' place of the product. Since there are no thou- 
sands in the multiplicaud. nor any to be carried, we put a 
cypher in that denomination of tho product, to keep any 
significant iigures that follow, in their proper places. 



i 



i 



MULTIPLICATION. 



5U 



41. Reason of I.— Tho multiplier ia to ))o pluccd under that 
dcnominfiMon of the multiplicaud to wliicli it belouRs; sinco- 
tliere is tJien no doubt of its vhIuo. Sometimes it is necessary 
(0 add cypiiers in putting down the muiti|il:er ; thu.s. 
EXAMPI.E 1.— 478 multiplied by 2 liundred— 
47H multiplicand. 
200 multiplier. 



Example 2.-539 multiplied by 3 ten- thousandths— 
68'J • multiplicand. 

0-0003 multiplier. 



Reason of II.— It ia similar to that given for tlio separatinff 
line in subtraction [10]. ^ e 

Reason ov III.— Wlien tho multiplicand exceeds a certain 
amount, the powers of the mind are too limited to allow us 
to multiply it at once ; we therefore multiply its parts, in suc- 
CQSsiun, un.l add the results as wo proceed. It is clear that 
tho sum of the products of the parts by the muliinlior, is equal 
to the product of tho sum of tlie parts by the same multi- 
plier :— tlius, 537x8 is evidently equal to 500 x8-f;;0x 84-7x8 
For multiplying all the parts, is multiplying the wliole ; since 
the whole is equal to the sum of all its parts. 

We begin at the rigl.^ hand side to avoid the necessity of 
athnimrds adding together the subordinate products Thus 
taking the example given above ; were wo to begin at the left 
liand, the process would be — 
897351 
4 

3200000=800000x4 

360000= 90000X4 

28000^ 7000x4 

1200= 300x4 

200= 50X4 

4= 1x4 



3589404=8um of products. 

iV^Z^ri °^ jy:~^'r'^,*^'® ''""^ ^'' ^^''"^^ "^f "'« fourth part of 
the rule for addition [9]; the product of the multipl/er and 
any denomination of the multiplicand, being equivalent to the 
bum of a colur. n m addition. It is easy to change the o-iveu 
ixainp e to an .xercise in addition; for 807851 x I, is theime 



thing as 



897351 
897331 
897351 
897351 

3589404 



m 



J 



fl 



60 



hlULTirUCATlOS. 




RKABopr OF y.-It follows, that tho Inflt pro^liict h to be eot 
<lown in ful; tor tlie tens it contains will not bo incroaseU : 
they in«y, tlioroloro, bo sot down at once. 

This riilo includca all casos in wlilnli tho ahsolii/e 
value ^ of the di^'its in the luultiplior d.x-s not excoea 
, 12. Their rcdativo value is not niatori-ii ; for it is as 
easy to multiply by 2 thousands as by 2 units. 

42. To prove multiplication, wluni tho mnltiplier dooa 
not exceed 12. Multiply the multiplionnd by th(> mul- 
tiplior, minus one ; and add the multlplicjind to the pro- 
duct. Tho sum should bo the same as the product of 
tho multiplicand and multiplier. 

Example.— Multiply G432 by 7, and prove tho i-viult. 
C432 multiplicand. 
6=7 (the multiplier) ~1 

6432 3S502 multiplicand xO. 

7(=C+1) 0432 multiplicand Xl. 

45024 = ' 45024multipllcandmultip]iodby 0,1=7. 

We have multiplied by 0, and by 1, and adtlod the results ; 
but SIX times the multiplicand, plus once the multiplicand, 
IS equal to seven times tho multiplicand. What we obtain 
from the two processes snould be the same, for we Wve 
merely used two methods of doing one thino-. 

EXERCISES FOR THE PUPIL. 



Multiply 
Bj 


(1) 

76762 
2 

(5) 
763452 
6 

(9) 
866342 
11 


(2) 
67450 
2 

(6) 
456769 
7 


(3) 
78976 
6 

(7) 
854709 
8 


(4) 

57340 
6 






Multiply 
By 


(8N 

45678f 
? 










Multiply 

By 


(10) 
788679 
12 


(H) 

476387^ 
11 


(12) 
fa>t29763 
12 













MULTIPLICATION. 



61 



5t ?fl to be Bot 
)o incronsed : 

;ho ahsolnfe 
not (jxcood 
for it is as 

Itiplier dooa 
ly the HI al- 
io tlio pro- 
product of 

I'Viult. 



th» reanlts ; 
ultiplican<l, 
i we obtain 
3r we Wve 



(4) 

57040 
6 



43. To Multiply when the Quantities contain Cyphers, 
or Dmrnals. — Slie rules alroady given aro applicable ; 
those which follow aro consetjuonces of them. 

When thoro arc cyplicrs at the cud of tho multipli- 
cand (cyphers in tho middle of it, Lavo been already 
noticed [40])— 

Rule. — Multiply as if there were none, and add to tho 

product as many cyphers as have boon neglected. For 

Tho greater tho quantity multiplied, tho grontor ought to 
be tho product. 

Example. -Multiply 5G000 by 4. 

5C00O 
4 



224000 



4 timoa imita in tho fourth place from the decimal point, 

arc evidently 24 ixnits in the same place ; — that is, 2 in tha 

fiflh place, to be carried, and 4 in the fourth^ to be set down. 

That wo may leave no doubt of the 4 being in tho fourth 

}>lace of" tho ;or()duct, we put three cyphers to tho rij^ht 
land. 4 times G are 20, and tho 2 to be carried, make 22. 

44. If tho multiplier contains cyphers — 

Rule. — Multiply as if there were none^ and add to 

the product as many cyphers as have been neglected. 

Tho greater the multiplier, tho greater the number of times 
the multiplicand is added to itself; and, therefore, the greater 
the product. 

ExAMPLK.— Multiply 507 by 200. 

5G7 
200 



113400 
From what we have said [35], it follows that 200x7 is 
the same as 7x200 ; but 7 times 2 hundred are 14 hundred ; 
and, consequently, 200 times 7 are 14 hundred ;~that is, 1 
in tho fourth place, to be carried, and 4 in the third, to be set 
down. We add two cyphers, to show that the 4 is in the 
third place. 

45. If both multiplicand and multiplier contain 
cyphers — 

Rule. — Multiply as if there were none in either, and 
add to the product as many cyphers as are found in 
both. 

d2 



m 



62 



MULTirLICATiON. 



Each of the quantities to be multiplied adcla cyphers to tho 
product [43 and 44]. 

Example.- Miihiply 46000 hy 800. 

40000 
800 



50800000 



_ 8 times G thousand yrocld bo 48 
times six thousand ought to prod 



thousand 

number 



8 hundred 
100 times 



greater— or 48 hundred thousand ;— that is, 4 in the scvcnt/i 
place from the decimal point, to be carried, and 8 in tlie 

are required. 



But, 5 cyph 



xLxtk place, to be set down. 

to keep the 8 in the sixth place. After ascertaining the 
position of the first digit in the p.^duct— from what the 
pupil already knows— there cjin be no difficulty Avith tho 
other digits. 

46. When there are dechnal places in the multipli- 
cand — 

Rule. — ^i\Iultiply as if there were none, and remove the 
product (by nieails of the decininl point) so many places 
to the right as there have been docuuals neglected. 

Tlie smaller the quantity multiplied, the loss the product 

Example.— Multiply 5-07 by 4. 

5-67 
4 



22-08 

4 times 7 hundredths are 28 hundreths :— or 2 tcntlis, to 
bo carried, and 8 hundredth « — or 8 in the second place, to 
the right of tho decimal point, to be set down. 4 times 6 
tenths are 24 tenths, which, with the 2 tenths to be carried, 
make 20 tenths ; — or 2 units to be carried, and G tenths to 
bo set down. To show that tlie re[>rosents tenths, we put 
the decimal point to tho left of it. 4 times 5 units are 20 
wiits, wliicli, with the 2 to ])e carried, make 22 units. 

47. When there are decimal:? in the multiplier — 
Rule. — Multiply as if there wore none, and remove 

the product so many places to the right as there are 

decimals in the multiplier. 

The smaller tho quantity by which we multiply, the less 
must be the rwult. 



)hers to tlio 



: 8 hundred 
100 times 
the seucnt/i 
d 8 in tlie 
e required, 
iiining the 
1 what tlie 
7 Avith tho 

3 muUipli- 

•cmove the 
any places 
cted. 

product 



I tcntlis, to 
d place, to 
4 times 6 
he carried, 
3 tenths to 
the, we put 
nJtH are 20 
nits. 

ier — 
id remove 
there are 

y, the less 






Example.- 



MULTIPLICATION, 

-Muiaply 5Go by -07 
503 
007 



63 



39-41 
3 multiplied hj 7 hundredths, is the same [351 as 7 hun- 
dredths multiplied by 3 ; whioh is equal to 21 hundredths : — 
or 2 tenths to be carried, and 1 hundredth — or 1 in the 
second place to the right of the decimal point, to be set down. 
Of course the 4, derived from the next product, must be 07ie 
place from the decimal point, «;c, 

48. When there are decimals in both multiplicand 
and multiplier — 

Rule. — Multiply as if there were none, and move 
the product so many places to the right as there are 
decimals in both. 

In this case the product is diminished, by the emallnesB of 
both multiplicand and multiplier. 

Example 1.— Multiply 56-3 by -08. * 

56-3 
•08 



4-504 



8 times 3 tenths are 2*4 [46] ; consequently a quantity 
one hundred times less than o — or -08, multiplied by three- 
tenths, vrill give a quantity one hvmdred times less than 2-4— 
or -024 ; that is, 4 in the third place from the decimal point, 
to be set dowTi, and 2 in the second place, to be carried. 

Example 2.— Multiply 5-63 by 0- 00005. 

5-63 
0-00005 



0-0002815 

49. When there are decimals in the multiplicand, and 
cyphers in the multiplier 5 or the contrary — 

Rule. — Multiply as if there were neither cyphers 
nor decimals ; then, if the decimals exceed the cyphers, 
move the product so many places to the right as will be 
equal to the excess ; but if the cyphers exceed the deci- 
mals, move it so many places to the kft as will be 
equal to the excess. ' 

The cyphers move the product to the left, the decimals to 
the right ; the effect of both together, therefore, will be equal 
to the difference of their separate effects. 




64 



MULTIPLICATION. 



PI 



ExAxMPLE 1.— Multiply 4600 bv "06 

4000 ^ 

006 2 cyphers and 2 decimals J excess -=0 



276 



Example 2.— Multiply 47-63 by 300. 
47-63 "^ 

300 



2 decimals and 2 cyphers; excess =0. 



14289 



Example 3.— Multiply 85-2 by 7000. 

_J^^ 1 decimal and 3 cyphers ; exce8fl=2 oji>Men 

596400 

Example 4.— Multiply 578-36 by 20. 
578-35 

^^ _ 2 decimals and 1 cypher; excess =1 decimal. 
11567-2 



Multiply 
By 



EXERCISES FOR THE PUPIL 

(13) (14) 

48960 76460 

5 9 



(15) 
678000 
8 



(16) 
57d00 
6 



Multiply 
By 



(17) 
7463 
80 



(18) 

770967 
900 



(19) 
147005 
4000 



(20) 

661*76748 
SOOOO 



Multiply 



(21) 
743560 
800 



(22) 
534900 
SOOOO 



(23) 
60000 
300 



(24) 

86000 

6000 




Slultiply 
By 



(25) 

62736 
o 



(26) 

8 -7563 
4 



(27) 
•21875 



(28) 
0-0007 



8 



MULTIPLICATION. 



05 



ss — 



Multiply 

By 



(29) 

5G341 

0-0003 



(30) 

85G37 
0-005 



(31) 

721*58 

0-0007 



(32) 
217G-38 
0-06 



3=0. 



2oyj!i.ewi 



L decimal. 



(16) 

67000 
6 



(30) 

o6{t76748 
30000 



(24) 

86000 

5000 



(28) 
0-0007 



8 




Multiply 

By 



(83) 
875-432 
0-04 



(34) 
78000 
0-3 



(35) 
51-721 
GOOO- 



(36) 
3*^ 
0-00007 

•00224 



In the last example we are obliged to add cyphers to the 
product, to make up the required number of decimal places. 

50. When both multiplicand and multiplier exceed 
12— 

KuLE.— I. riace the digits of the multiplier under 
those denominations of the multiplicand to which they 
belong. 

II. Put a line under the multiplier, to separate 'u from 
the j)roduct. 

III. Multiply the multiplicand, and eack part of the 
multiplier (by the preceding rule [39]), beginning With 
the digit at the right hand, and taking care to move the 
product of the multiplicand and each sncce.ssive digit 
of the multiplier, so mnny places more to the left, than 
the preceding pi-oduct, as the digit of the multiplier 
winch produces it is more to the loft tlian the signifi- 
cant figure by which we have kusi multiplied. 

IV. Add together all tlie products; and their sum 
will be tlie product of the multiplicand and multiplier. 

51. ExAMPLi:.— Multiply 5634 by 8073. 
5034 
8)73 



lG002=prodact by ?,. 

39438 =pro(lact l)y 70. 

45072 =product by 8000. 

45483282=product Ijy 8073. 

The product of the nuiUiplicand by 3, requires no e^i^ 
nation. ^ 




66 



MULTIPLICATION. 




7 tens times 4, or [35] 4 times 7 tens arc 28 tens : — 2 hun- 
dreds, to be carried, and 8 tens (8 in the second place from 
the decimal point) to be set down, &c. 8000 times 4, or 4 
times 8000, are 32 thousand : — or 3 tens of thousands to be 
carried, and 2 thousands (2 in the fourth place) to be set 
down, &c. It is unnecessary to add cyphers, to show the 
values of the first digits of the different products ; as they 
are sufficiently indicated by the digits above. The products 
by 3, by 70, and by 8000, are added together in the ordic-y 
way. 

52. Reasons of I. and II. — They are the same as those 
given for corresponding parts of tlie preceding rule [41]. 

IIEASON OF III. — We are obliged to multiply successwely 
by the parts of the multiplier ; since wo cannot multiply by 
the whole at once. 

xlEAsoisr OF IV. — The sum of the products of the multipli- 
cand by the parts of the multiplier, is evidently equal to the 
■product of the multiplicand by the wliole multiplier ; for, in 
the example just given, 5634 X 8073 = 5684 X 8000 -f- 70 -f 8= 
[34] 5034 X 8000+5634x70-1-5634x3. Besides [35], we may 
consider the multiplicand as multiplier, and the multiplier as 
nmltiplicand ; then, observing the rule would be the same 
thing as multiplying the new multiplier into the diiFerent 
parts of the new multiplicand 5 which, we have already seen 
[41], is the same as multiplying tlie whole multiplicand by 
the multiplier. The example, just given, would become 
8073X5634. 

8073 new multiplicand 
5684 new multiplier. 



We are to multiply 3, the first digit of the multiplicand, by 
6634, the multiplier; then to multiply 7 (tens), the second 
digit of the multiplicand, by the multiplier ; &c. When the 
multiplier was small, we could add the different productti as 
we proceeded; but we now require a separate addition, — whicii, 
however, does not affci the nature, nor the reasons of the 
process. 

53. To p'ove multiplication, when the multipliei ex- 
ceeds 12 — 

EuLE. — Multiply the multiplier by the multiplicand ; 
and the product ought to be the same as that of the 
multiplicand by the multiplier [35] . It is evident, that 
we could not avail ourselves of this mode of proof, in tho 
last rule (b[42j ; as it would have supposed the pupil to 
be then able t-o multiply by a quantity greater than 12 



th 
de 
lei 
in 

of 



Til 





ini 


7- 


ha 
or 


'■/ 


aol 


« 


th( 


'4 





Tal 

1 
bc^ 

1 



'I 



3 : — 2 hun- 
3laco from 
lies 4, or 4 
iauda to be 
) to be set 
show the 
i] as they 
e products 
e ordic-y 

e as those 
[41]. 

uccessively 
aultiply by 

e multipli- 
jual to the 
er ; for, in 
f 70+8= 
)], we may 
ultiplier as 
the same 
e diiFerent 
ready seen 
plicand by 
Id become 



plicand, by 
the second 
When the 
)roductt as 
n, — whici., 
ions of the 

tipliei ex- 

tiplicand ; 
lat of the 
dent, that 
oof, in tho 
B pupil to 
• than 12 



MULTIPLICATIOINr. 



67 



mR 



54. We may prove multiplication by what is called 
" casting out the nines." 

Rule.— Cast the nines from the sum of the digits of 
the multiplicand and multiplier ; multiply ti-e remain- 
ders, and cast the nines from the product :— what is now 
left should be the same as what is obtained, by cast- 
ing the nines, out of the sum of the digits of the product 
of the multiplicand by the multiplier. 

I^.XAMPLE 1. — Let the quantities multiplied be 942G and 

'I'aking the nines from 9426, we get 3 us remainder. 
And from 3785, we get 5. 

47130 

75408 3x5=15, from which 9 

C5982 beino; taken, 

28278 • 6 are left. 






Tiiking the nines from 35077410, 6 are left. 

The remainders l)e!ng equal, we are to presume tlie 
multiplication is correct. Tlxe same result, however, would 
liave been obtained, even if we had misplaced digits, added 
or omitted cyphers, or fallei. into errors which had counter- 
acted each other : — with ordinary care, however, none of 
these is likely to occur. 

ExAMPr.K 2.— Let the numbers be 70542 and 8436. 
T:>„king the :aincs from 76542, the remainder is G. 



Taking them from 



8436, it is 3. 



459252 

229626" 6x3=18, the 

306108 remainder from which is 0. 
612336 



Taking the nines from 645708312 also, the remainder is 0. 

Tho remainders being the same, the multiplication may 
be considered right. 

Example 3.— Lot the numbers be 403 and 54. ^ 
From 463, the remainder is 4. 
From 54, ',-■ ' - 



1852 4x''-=0 from which the remainder is 0, 

23 15 

From 2bd02 the remainder is 0. 



u 



kl 



% 



68 



MULTIPLICATION. 



Tlie remainder being in each case 0, wo arc to suppose 
that the multiplication is correctly performed. 

This proof applies whatever be the position of the 
decimal point in either of the given numbers. 

55. To understand this rule, it must be known that 

a number, from which 9 is taken as often as possible, 

will leave the same remainder as will be obtained if 9 

be taken as often as possible from the Bum of its di.^its " 

Since the pupil is not supposed, as yet, to have learned 
divinon, he cannot use that rule for the purpose of 
casting out the nines ;- nevertheless, he can easily 
ellect this object. •^ 

K !f, o'" f^7^'\ ^^""^^'er be 5C3. The sum of its digits is 
+,.+'^' ^hile the nvimber itself is 500-fG0+3. 
First, to take 9 as often as possible from the sum of //,« 

(hgrls. 5 and 6 aro'll ; from which, 9 being taken, 2 are 

loit. ^ and 6 are 5, which, not containing 9, is to be set 

down as the nmainder. 
Next, to ta^o 9 as often as possible from the mmbcr itself.. 

503^=500 + 00+3=5 xl00+Gxl0+3=5x9iq^+Gx 
9+1+3,= (if we remove the vinculum [34]), 5x99+5+ 
Ox.i+b+3 But any nnmber of niacs, will be found to he 
f tie product of the same number of ones by 9 .-—thus 999— 
111X9; 99=11x9; and 9=lx9._Hence 5x99 express;^ 
a certain number of nines-being 5x11x9 ; it may, there- 
lure, be cast out; and for a similar reason, Gx9: after wliich 
there will then be left 5+G+3-from w'hic^i the luues are 
still to be rejected; but, as this is the sum of the dibits we 
must, in casting the nines out of it, obtain the same remain- 
aer as before. Consequently "we get the same remainder 
whether we cast the mnes out of the number itself, or out 
of the sum of its digits." 

Neither the above, nor the following reasoning can 
offer any difficulty to the pupil who has made himself 
as fainiiar with the use of the signs as he ought :- 
they will both, on the contrary, serve to show how much 
simpbcity, is derived from the u«e of characters express- 
ing, not only quantities, but processes ; for, by nieanj 
ot such characters, a long series of argumentation mav 
be seen, as it were, at a single glance. 

5G "Costing the nines from the factors, n-.iripiyina- tU 
resulting remainders, and casting tlie nines from thiH product, 



MULTIPLICATION. 



09 



to suppose 
ion of the 

nown that 
8 possible, 
aincd if 9 
its digits." 
ve learned 
urpose of 
3aa easily 



;s digits is 



sum of it.<i 

ken, 2 are 

to he set 

mhcr itself: 

XOO-fS-f- 
und to be 
hus 999= 
expresses 
I ay, tlioro- 
tov M-liich, 
nines are 
digits, we 
fe remain- 
■emainder 
elf, or out 

ning can 
himself 
)ught : — 
3W much 
eypress- 
y means 
ion ma;i 

yinj>- tli<» 
product, 



will Iftave the same remainder, as if the ninps were east from 
the product of the factors," — provided the multiplication 
has boon rightly performed. 

To bhuw tliis, set down the quantities, and take away the 
nines, as before. Let the factors be 573x464. 

Casting the nines from 5-J-74-3 (which we have just seen 
is the same as casting the nines from 573), wo obtain 6 as 
remainder. Casting the nines from 4-f-G-j-4, we get 5 as 
remaiiuler. Multiplying 6 and 5 we o btain 30 as product ; 
which, being equal to 3x10=3x94-1=3x0-4-3, will, when 
the nines are taken away, give 3 as remainder. 

Wii can show that 3 will be the remainder, also, if we 
cast the nines from the product of the factors ; — which ia 
clFected by sotting down this product ; and taking, in suc- 
ecssion, quantities that are equal to it — as follows, 

573x404 (the product of the factor8)= 

SxBO+T xlO+S X 4 x 1004-6 xl0-}-4= 



5x99-fl-f7x9+l4-3 X 4 x 994-14-6 x94-l4-4= » 
5x994-54-7x94-7+3 X 4 x 994-44-6 x94-6-|-4. 

5x09, as we have seen [55], expresses a number of nines; 
it will continue to do so, when multiplied by all the quan- 
tiiies under the second vinculum, and is, therefore, to be 
cast out; and, for the same reason, 7x9. 4x99 expresses 
a number of nines ; it will continue to do so when multiplied 
by the quantities under the first vinculum, and is, therefore, 
to be cast out; and, for the same reason, 6x9. There will 
then be left, only 54-74-3 X4-1-64-4, — from which the nines 
are still to be (Tast out, the remainders to be multiplied together, 
and the nines to bo cast from their product ; — but we have 
done all this already, and obtained 3, as the remainder. 

EXEnCISES FOR THE PVPII.. 



Multiply 
By 


(37) 

765 

^ 765 


(38) 

732 

456 


(39) 

997 
845 


(40) 

767 
347 


I'jtroducts * 










Multiply 
By 


(41) 
657 
789 


(42) 

456 
791 


(43) 
767 

789 


(44) 

745 

741 


rroducts 


1 


■ 







70 



MULTIPLICATrON. 



.«n J' u! r "" ^yP^ors, or decimals in the multinli, 
cand, rnultrpher, or botli ; the same rules apply as when 
the niultipliar does not exceed 12 [43, &c.] 



(1) 
4600 
67 



(2) 
2784 
620 



(3) (4) 

32-68 7856 
26- 0-32 



(5) (6) 

87-96 482000 
220- 0-37 



2G2200 1726080 849-68 2618-92 19351-2 178340~" 
Contractions in Multiplication. 
r-.,f;ioT^^° '!i'' °«* necessary to have as many deci- 
aTd\&"lier-' ^"'"'*' ^^ "^ ^" ^^^^^ -^^^^P^--^^ 

un?erM;r^r'''? *i' multiplier, putting its xmlis^ place 
oZ? I' 1^ ""^i ^^, *^^^* denomination in the multipli- 
cand, which IS the lowest of the required product. ^ 
^-.^'iW ^^.^^«h digit of the multiplier, beginninff 
With the denomination over it in the multipl cand ; Tuf 
addmg wha would have been obtained, on multip yW 
the precedmg digit of the multiplicand-unity, if^the 
nmnber obtained would be between 5 and 15 - 2 if 
between 15 and 25 ; 3, if between 25 and 35 ; &c 

frt^Kl }7^^^ ^T^'''''^'''''^ ^^ *^^« products, aripinff 
from the diflFerent digits of the multiplicand, stand ia 
the same vertical column. ' 

Add up all the products for the total product; from 
which cut off the reqmred number of decimal places. 

59. Example 1.— Multiply 5G784 bv 97324 sn oa +« 
have four decimals in the priuct ^^-^^^24, so as to 

Short Method. Ordinary M.thod. 



56784 
42379 

511056 

39749 

1703 

113 

22 

55^2643 



5-67r t 
9-7324 

22i7136 
1131568 
1703i.")2 
39748 8 
51105G 



'4 



55 -2044 601 (] 



i j^LM. 



MULTIPLICATION. 



71 



9 in the multiplier, expresses units ; it is therefore put 

ander tho/o«r</i decimal pliioo ol'tho multiplicand— that being 
tho place of the lowest decimal required m the product. 

In multiplying by each succeeding digit of tho multiplier, 
we neglect an additional digit of the multiplicand; because, 
as tlie multiplier decreases, the number multiplied must in- 
crease—to keep the lowest denomination of the ditterent pro- 
ducts, tho same as the lowest denomination required in tlie 
total product. In the example given, 7 (the second digit of 
the multiplier) multiplied by 8 (the second digit of the mul- 
tiplicand), will evidently produce the same denomination as 9 
(one denomination higlier than tbo 7), nniltiplied by 4 (one 
denomination lower than the 8). Were we to multiply tho 
lowest denomination of the multiplicand by 7, we should get 
[4(5] a result in iha Jift/i place to the right of the decanal point ; 
which is a denomination supposed to bo, in the present in- 
stance, too inconsiderable for notice— since we are to havo 
only four decimals in the product. But we add unity for 
evt'.ry ten that would arise, from the multipl cation of an ad(". 
tional digit of the multiplicand ; since every such ten consti- 
tutes one, in tlie lowest denomination of the required product. 
When the multiplication of an additional digit of the inulti- 
pliciind would give more than 5, ..ud less than 15 ; it is nearer 
to the truth, to suppose we have 10, than either 0, or 20 ; and 
thereiore it is more correct ta^add 1, than either 0, or 2 When 
It would give more than 15, and less than 25, it is nearer to 
the truth to suppose we havo 20, than either 10, or SO ; and, 
therefore it is more correct to add 2, than 1, or 3; &c Wa 
may consider 5 either as 0, or 10 ; 15 eil/ier as 10, or 20 ; &c. 

On inspecting the re.sults obtained by the abridged, 
find ordinary methods, the difference is perceived to bo 
inconsiderable. When greater accuracy is desired, wo 
should proceed, as if we intended to havo more decimals 
in tho product, and afterwards reject those which are 
unnecessary. 

EvAMPLE 2.— Multiply 87653^ by -5704, so as to hav» 
6 decimal places. 

Mt 8-76532 

^ 4G75 

4383 

G13 

62 

3 

6051 



,s^i 



lil'ri 



:_. ai 31! :U 



72 



MULTIPLICATION. 



Tliere arc no units in tho multipljpr; but, as the rule 
dirocts, wo put its units' place under tli^ third decimal place 
ol the nmltipHcjuii. In multipivlng by 4, since there is no 
di;,'it over it iu Lho multiplicand, we merely set down what 
would have resulted from multiplying tho precodiu«r dono- 
mmatlon of the multiplicand. ° ^ a 

Example 3.--M.dtiply -4737 by -6731 so as to have 
docnnal places in tiio product. 

•47370 
137G 



284220 

33159 

1421 

47 

•318847 



_ Na have put ^.he units' place of the multiplier under tho 
suth decimal place of tho multiplicand, adding a cypher, or 
su^iposing it to be added. 

Example 4.— Multiply 84G732 by -0050, sc as to have 
lour decimal places. , 

84- 0732 
G5 



4234 
508 



•4742 



Example S.—Multiply -23257 by -243, so as to have four 
decimal places. 

23257 
342 



465 
93 

•05G5 

AVe are obliged to place a cypher in the product, to mako 
up the required number of decimals, 

00. To multiply by a Composite Number — 
KuLis.— Multiply, successively, by its factors. 



I 



8 the rule 
inial ^lacu 
tlioro i- no 
ovnx what 
iiug dcno- 

to have 6 



[inder the 
sypher, or 

} to have 



liave four 



to malcG 



i 



MULTIPLl ATION. 



73 



EyAMPT.K— Multiply 732 by 90. 90=8x12' Lhotoforo 



732x1"' 



732 x8x 12. 

732 
8 



[35 



I 



5850, product by 8. 
12 



70272, pi .act by 8x12, or 90. 

If we multiply by 8 only, Ave multiply by a quantity 12 
tinios too Hinall ; ami, therefore, tlie product will bo 12 times 
loHH than it slioul ' We rectify tliis, by making the product 
J J times greater— ...at ia, we multiply it by 12. 

fU. When the multiplior is not exactly a Composltcf 
Nui!i))or — 

lluLi:. — Multiply by the factors of the nearest com- 
posite ; and add to, or subtract from the last product, 
>s{) many times tlie multiplic \ as the assumed compo- 
site is le.ss or greater than the given multi2)licr 

Example 1. — Multiply 927 by ^7. 

87 = 7 X 124-3 ; therefore 927 X 87 = 927 X 7x 12+15 = 
927x7x12 + 927x3. [34]. 

927 
7 



0489: 
12 



:927 X 7. 



77808 = 027x7x12. 
2781 = 927x3. 



80049 = 927 X 7 x 12 + 927 X 3, or 927 x 87. 
If we multiply only by 84 (7 X 12), we take the number to 
bo multiplied o times less than we ought ; this is rectified, by 
adding 3 times the multiplicand. 

ExAMPLK 2.— Multiply 432 b y 79. 79 = 81-2=9 x 9-2; 

thoroforo 432 X 79=432 x 9 X 9-2=432 x 9 x9-432x 2! 
432 
9 



3888 = 432x9. 
9 



34992=432x9x9. 
804=432x2. 



34128=432 x 9 x 9-432 x 2, or 432 x 79. 








IMAGE EVALUATION 
TEST TARGET (MT-3) 




1.0 



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2.2 



1^ illlU 

Ik ..ill 

IIIIIM 
u mil 1.6 



^. 



% 



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/^ 



* 



'c^l 






^^ > 



^ cf 



^^#^5^' 



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HiotograDhic 

Scieices 
Corporation 



23 WEST MAIN STREET 

WEBSTER, N.Y. 14580 

(716) 872-4503 





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74 



MULTIPLICA7I0K. 



fli nifn K '^ r^° ^^ ^}'. *''° composite number, we have taken 
Jh7product ^ subtracting twice the multiplicand from 

62. This method is particularly convenient, when the 
muKipher consists of nines. 

To Multiply by any Number of Nines,— 
liuLE.— Kemove the decimal point of the multipli, 
cand so many places to the right (by adding cyphers if 
necessary) as there are nines in the multiplier • and 
subtract the multiplicand from the result. ' 

.ExAMPLK.— Multiply 7347 by 999 ' 

^ 7347 X 909 = 7347000-7347=7339053. 
We, in such a case, merely muliinlv hv the ««»* i.* ^, 
convenient composite number,'and "Sra^I the muHiputnJ 
:xr;je^Ju«tS;4r ^^^- ^^ *- ^^^en; thuftrh'a 

7347x999=7347xi000-l=:7347000-7347=7339653. 

63 We may sometin.cs abridge multiplication hv 
oonsidermg a part or parts of the multiplier as pro- 
duced Dy multiplication of one or more other parts. 

ExAMPLK -Multiply 57839208 by 62421648. The mnl. 
tipUer may be divided as follows :-0, 24, 216 and 48 

24 = 6x4 
216 = 24x9 
48 = 24x2 
57830268, iimltiplicand 
62421648 , multiplier. 

''S??oP3^ • = P''0(^"«t1>y 0(60000000). 
I.ih8142432 : : product by 24 (2400000^ 
12493281888 : product by 216721600/ 
2776284864 product by 48. ^" 



3610422427073664 product by 62421048. 
,1 ^i'lP^o^?"^^ by 6 Avhen multiplied by 4 will dve the nrn 
trn\ Y 24; the product by 24, Multiplied ^9, iill eive^th; 
product by 216-and, multiplied by 2. tlie prodactTy 4!! 

fJt' Ji-'-f ''''.'\^^ !^^ difficulty in finding the places of 
the first digits the difi-erent products. For when thoro 
are neither cyphers nor decimals in the multiplicand— 
ndthorTiS n;ultiplication, we may suppose that there are 
neithei [4b, &c.]— the lowest douomination of each pro- 



MULTII'LICATION. 



75 



duct, will be the same as the lowest denomination of ths 
multiplier that produced it ; — thus 12 units multiplied 
by 4 units will give 48 units ; 14 units m.ultiplied by 4 
tens will give 56 tens ; 124 units midtiplied by 35 units 
will be 4340 units, &c. ; and, therefore, the beginning of 
each product — if a significant figure — must stand under 
the lowest digit of the multiplier from which it arises. 
When the process is finished, cyphers or decimals, if 
necessary, may be added, according to the rules already 
given. -^ 

The vertical dotted lines show that the places of the lowest 
digits of the respective multipliers, or those parts into which 
the whole multiplier has been divided, and the lowest digits 
of their resulting products are — as they ought to be — of the 
same denomination. 

48 being of the denomination units, when multiplied into 
8 units, will produce units; the first digit, therefoi'e, of the 
product by 45 is in the units' place. 216, being of the deno- 
mination himdreds when multiplied into units will give hun- 
dreds ; hence the first digit of the product by 216 will be in 
the hundreds' place, &c. The parts into which the multi- 
plier is divided are, in reality, 

COOOOOOO 



2400000 

• 21600 

48 



.=62421648, the whole multiplier. 



We shall give other contractions in multiplication 
hereafter, at the proper time. 



EXERCISES. 



45. 745X456^:=339720. 

46. 476X767=365092. 

47. 345X579=199765 

48. 476X479=228004. 

49. 897X979=878163. 

60. 4 •59X706=3235 -95. 

61. 767X407=312169. 

52. -467 X- 606= '276942. 

53. 700X810=567000. 

54. 670X910=009700. 

55. 910X870=791700. 
66. 5001-4x70=350098. 
57. 64 -001X40=2560 -04. 
68. 91009X79=7189711. 
59. 40170X80=3213600. 



60. 707X604=427028. 

61. 777 X •407=318-239. 

62. 7407X4404=32620428. 

63. 6767X1307=7537469. 

64. 67 •74X -1706=11 -556444 

65. 4567X2002=9143134. 

66. 7-767x301-2=2339-4204 

67. 9600X7100=68160000. 

68. 7800X9100=70980000. 

69. 6700X6700=44890000. 

70. 5000X7600^--38000000. 

71. 70814x90l07=63808-37098. 

72. 97001X70706=7440658706. 

73. 98400X07407=6295813800. 

74. -56007x45070=25242-35490 



70 



WULTIPLrCATION. 



aolmtir"^''""'"^' '" ^1395; a pound being 

7b. In 2480 pence how many farthings : four far- 
thmgsbemgaponny.. ° A«. 9920. 

I. J f;r'87lhunSf: p ■="' " ^■''"'"^' ''- ™r "f;,^; 

'TO IT 1° .1, Ans. 1479. 

ton f "''' ' ^^^ ^^"^ ^^ ^^***^^ ^°«<^ ''^ ^25 a 

'TO 71? 1 /. 7l7w, 012;') 

will 119 n^ ^T f /°^ *^^°S cost 4 pence, how much 

80 llZ " '"'* • • ^^"- 448 pence. 

80 Row njany pence in 100 pieces of coin, each of 

which 18 worth 57 pence ? a,,, r'inn 

CI TT^,„ f^^^^ jins. 5700 pence. 

t«inL RQ ^^^ypUons in 264 hogsheads, each con- 
taining 63 gallons .? ° L,, Tp,'" 

82. If the interest of ^1 be £0-05, how much wi5i 
be the intarest of ^6376 ? ' Ans £l 8 8 

cosll" "^ ''''^ """^'"^^ '°'* ^^'^^' ""^^^ ^^^^ 973 such 
84. It has been computed that tho gold, silver and 
brass expended in building the temple%f So omon at 
Jerusalem amounted in value to ^£6904822500 of our 
money ; how many pence are there in this sum oZ 

Ts'Stir"''- . 1 ^-165VV57SC 

85 I he followmg are the lengths of a degree of the 

?em •1?):86'6'- 't"?"-^ ^"""^'^ 60480-2\ThoLin 
. n ' ,^^^^^^ m I»<iia ; 60759-4 in France • fiOS^fi-fi 
m England; and 60952-4 in Lapland. 6 feet beln' a 

st^Yt'^ r"^ ^''' ^^ ^^^^ «f ^'- above. P 1?.; 
362b8l2 m Peru; 362919-6 in India; 364556-4 in 
France ;r^650_l 9-6 in England ; and 365714-4 in Lapland 

86 The width of the Menai bridge between ?he 
points of suspension is 560 feet ; and th? weigtrbeLL 
these two points 489 tons. 12 inches bein/a foot and 
2240 pounds a ton, how many inches in°the fomTr 
and pounds m the latter ? loimer, 

87 Th..n ^"1' ^^2^.i?«^es, and 1095360 pounds. 
87 There are two minims to a semibreve • two 

crotchets to a minim ; two quavers to a ciSeV- wo 
semiquavers to a quaver : and two demi-sem quavcrslo 
I rerSre^P ^^^ demi-semiquavers i ~^ 

Ans. 221 



M' 



M ULTI PLICA n ON. 



77 



88. 32,000 seeds have been counted in a single poppy • 
how many would be found in 297 of these ? Ans. y50400o! 

89 9,344,000 eggs have been found in a single cod 
Lsh J how many would there be in 35 such ? 

^^ „„ , Ans. 327040000. 

^ 65 When the pupil is ftimiliar with multiplication, 
m workmg, for instance, the following example, 
897351, multiplicand. 

4, multiplier. 

3589404, product. 
He should say :— 4 (the product of 4 and 1), 20 (the pro- 
duct of 4 and 5), 14 (the product of 4 and 3 plus 2, to be 
earned), 29, 38, 35; at the same time putting down 
the units, and carryin^; the tens of each. 

QUESTIONS TO BE ANSWERED BY THE PUPIL. 

1. What is multiplication .? [24]. 

2. What are the multiplicand, multiplier, and nro- 
duct.? [24]. ^ ' ^ 

3. What are factors, and submultiples } [24] . 

^ 4. What is tlie difference between prime and compo-- 
site numbers [25] ; and between those which are prime 
and those which are composite to each other ? [27] . 

5. What is the measure, aliquot part, or submultiplo 
of a quantity ? [26] . 

6. What is a multiple .' [29]. 

7. What is a common measure ? [27T . 

8. What is meant by the greatest common measure > 
[28] . 

9. What is a c<9wmo% multiple .? [30]. 

10. What is meant by the kast common multinle > 
[30]. ^ • 

11. What are equimultiples .? [31]. 

12. Does the use of the multiplication table prevent 
multiplication from being a species of addition .? [33]. 

13. Who first constructed this table ? [33]. 

14. What is the sign used for multiplication ? [34]. 

15. How are quantities under the vinculum affecteJ!* 
by the sign of multiplication .? [34] . 

16. Show that quantities connected by the sign o/ 
multiplication may be read in any order ? [35] . 



WL^ai 



78 



iiirisioN. 



,0*';^',"'^ ""■■ '""llilJiw exceeds 12 > ran ' 

cccoAs I2t [Lt "'''' "'"" °»'^ *» nluUiplicHud 

*; J°' '^^^^''"■« tl"= rules when the mnltipUcand mnl 
' a^'^T-r; ""''.?""" "W'"''-^' ^ decimals ?[43;&Ti: 

2i' F."^i' °'""'P"''f<'? P^o^^i •' [42 and 53]. 

wtdgn?.Sl' ITsr""^'' ""^°"' ^"pp-»g » 

n»nfbo?„7del:rpLt¥^r5sV' '" ''"^ " '"^--^ 

^ J8. How may we multiply by any number of -Jnes > 
[esf: ^""^ '"" °"'"'P'i''^«en «ry briefly performed ? 
SIMPLE DIVISION. 

orde'eminatl""""'' "^ ^PJ*""-*' ^■" "»'- o^"^ 

calMTlTJ?'^""'™ '" *■"> °"' ''"w oft™ one number 
called the divtsor, is mnlaineil in, or can *, /^/J;,T ' 
another, termed the divi,i,^,l ■ .1 ^ i ™'«"/™« 

/-».//» is call d tt S;„T DKr™'f^P''"^T« 
u« t» tell, if a quantity be dWded into a cL?" '""f ' 

"'Xi^^'d^"-"' ^"^ "^ ti'lCt^ofrh" """'" 
wnen the divisor is not contained in the divid^n.? 

p.-ocea, would be required to'dLTrer^SyTub* 



vivitioa. 



70 



trading it~liow ofton 7 is contairiod in 8063495724 
jv^i.lo, a.s wo_«h.ll liud, the same thing can bo effoctcd 
by dicmmi, m less than a minute. 

08. Division is expressed by -f-, pLvced between the 

.v.dend and divisor; or by j.utting^he divisor und J 

the dividend, with a separating line between :— thus 

64-3=2, or-=.2 (road (3 divided by 3 is equal to 2) 

means, that if 6 is divided by 3, the quotient will be 2. 

<)9. When a quantity under the vinculum is to ).c 
divided, we must, on removing the vinculum, put the 
divisor under each of the terms connected by the .i^^u 
of addition, or subtraction, otherwise the value of wliat 

was to be divided will bo changed ;— thus 5T6^-^3=:r- 

6 7 ' ~ 

---!.____. f^^. ^^j j^^ j^^^ ^^.^..^^^^ ^1^^ ^j^^^^ ^^^^^^^ 

vvc divide all its parts. 

_ The line placed between the dividend and divisor occa- 
Monally assumes tlie place of a vinculum ; and there- 
ore, when the quantity to be divided is subtractive, it 
\uli sometimes be necessary to change ^' - * 

already directed [16]:— thus --+ ^^""^ 



the signs— as 
6 + 13—3: 



but 



27 
3 



^5— 6 + 9 ^ 27—15 + 6- 
3 



2 
-9 



2 



For when, aa 
in these cases, «// the terms are put under the vinculum 

"thf : '' v^f '^^? '"^^^^^*^''^ ^^S"^ -^'^ concern S- 
IS the same as if the vinculum were iSmoved alto-ether • 
and then the signs should be changed i../; .^a tJ 
what they must be considered to have been S tl e 
vmculum was alfiv^ed [16]. -^ ^ 

When quantities connected by the sjcrn of multinlien 
tion are to be divided, dividing Iny one° of trS.s' 
.nil be the_same as dividing the product ; thus, 5X10 X 
2o+5= - X lOX 25 ; for each is equal to 250. 

To Divide QuanlUits. 

70. When the divisor does not exceed 12, nor the 
dividend 12 times the divisor ' 



80 



DIVISIOIf. 



B.;le.~I imd by tlio mnltiplicntion taMo tho 
groutost number which, multiplied by the divisor wlU 
give a product that doo.s not exceed L diviS' 7\^ 
will be the quotient required ' 

number td'fh/r^" '^'' t^^''f *^« P^«^^«* «^ ^^"'^ 
anv w tlw i- • ''^''"' ', ''^'"« '^'^^^^ *^^" remainder, if 
any, with the divisor uuder it, and a lino between them 

aro^S T '''^ ^^-/""Itiplication tablollllo times G 

The total quotient is9+|,or9^; that is, ^=9^ 
^ji :^:S^t^XS^;rt^"^or, w^ean^^ct it 

ber'of^i^rThrcivTsox-Lnt;\i^ *?« f-*-* --- 

is, tlie greatest mull pie of 6 w], oh •i^'''"V'^^ dividend; that 
ber to°be divided The mnlTinH^^^ not exceed tho num- 
ducts of any two m^mhL '^ '^"^ ^^^'^^^ ''^'"^^■^ the pro- 

therefore iJirlblesu" to obt^i"^^!^^ ' "'^' ''''''^' ^^' '^"'^ 
must not exceed the dhiSend"^^ ^''^^^"^■«' ^•"'* 

leave a number equal to or '-olf' T° «".''t^''^«ted from it. 
hardly necessary loremrk t at .7 * i'""' '^'' ^^^^'•^«^- ^^ is 
been subtracter! .4 ofTon 1.' '.t -k'*" ?"'''"'' ^^"^'^ "«t have 
number equal to or ffreatel tlfan u'' ^''? i''« ^^'^^'^^"^ ^^ ^ 
quotient answer tleon^^Hnn / '^ r '''■'' ^'^^^ ' "'^^ ^^0^1'^ ti.e 
taken from the dividend °' ''"' "^'"'^ ^^'^ '^^^^^^^^ ««^^IJ bo 

anf^SenTf(4;r'JreSrnVL^ T'^^-V ^^ ^'^ ^^-«- 
remainder, what it i When^j. Je H';^ *'-"i'' ^' '^"^ 
J-eahty suppose tho dividend div eel iito IT'^^'f'' ""' '" 
these IS equal to the product nfThi- "^^ P'''^"*^ ^ «"« f>t' 



pie given, f =:^i±4=,l4 4 4 



72. When the divLsor does not exceed 12 bui M,« 
dividend exceeds 12 times the divisor-- ' '' 



DIVI.SION. 



tablo tlio 
li visor, will 
idend: this 

luct of this 
mainder, if 
yeen them. 

1 58; or, in 
)y G. 

10 times G 
Teforo, doea 
ible, that 9 
tly 6 is con- 
9 quotient; 
given num- 

4 

an effect it 



ateat num- 
dend; that 
I tho mini- 
's the pro- 
Is 12; iitid 
luire; this 
id from it, 
sor. It is 
1 not liave 
idend if a 
would tlie 
■ could bo 

le divisor 
be any 
er, we in 
s ; one of 
lent— and 
i between 
^s, hy the 
lie cxtiia,- 



bui the 



81 



liiiHBssias 



r 

10 

ino 



mg remainder, when there is one ind de /r/ '^I 
contain the divisor consido,. .V + ' ^"r"^"^) ^^^«« "ot 
the next lower and ^d, I ^ '''''' ^^ "'^'"^ ^^' 

the dociuml point. ' ^^'''' removed from 

"-Jis^et;:rit,"rit!t:; ^'^^ ^t^^-^ *'- 

[70]-,vith the div sor u nd l 7 i ^'^''"''^-^ ^^'""'^^'^^ 
l'otv.oen tliem ; o ^writ nt tl!' d" ' f'^"'-"'"^^' ^^"« 
quotient, proceU with t o^di! ""i^ ^^''"* ^" *''^ 

vonKundor'ten 1?^! t t nt^^'th'e ttt'l""'"' r'' 
mmathm ; proceed tliua Z^7fh • ^ ^''''''^'' ^*-^"'^- 
nntil it is so tr flin .7h. T- "'? '' "^ remainder, or 

iaconvenfence ° '^"^ ^' "'^^Slected without 

73. Ex.MrLE.--What is the quotient of 04450-^7 ^ 
Divisor 7)G445G dividend. ' ' 

1)208 quotient. 

i« greater thin GO t mm ?1 '' •'^'' '1* '^^^'"'''"'^^' ^^'^'^^''^ 
to he nut mle J T: \^'T '"' therefore, no di^it 

o;>evei, put a cyp icr m that place, since no digit 



B2 



DivisroN. 



l! 



drnfli4 '^ ■ • HI11L3 ^ JiiinUrods nvo (iT.ir.n,r i i i 

the 

go 

tens, which 

tens 

divid( 

in« 

phice of the quotioi.L, juiu cue 

quotient is fu,i„d to bo 9208 eaetly ; H.at is, tt^^ 92O8. 
b/fi J ^""'"'^ 2.-What is tl,e quotient of 72208, divided 

0)73208 

proceed with the division as followa-- ^ '' "'"-^ 

0)73208 

' ■ ., . , i22rf333,&c. 

L-onsiderinoj the 2 units inPt- f„ xi 

dend, as 20 Tenths, rpel-ceivetZ a" ".f ' '^ ^''^ ^^^^^- 
three tenths times and I^.tn if •''''" «'' '"t" t^^o'^ 

6 (=0 times 3 emhTf^t^f ' I ;f^^ ^ ^^"^^'^ ^^'''^ 

into .cThund^edths Vl:^^^ ^^-^ wn;^^ 



Ilin dooiiutil 
euMo, i»ro(Iiu!n 
of tlK.usatiil.s 
^iiiu.l» iilrciidy 
'• "so" into 
usiind times; 
I is Icsa tliau 
oos not Icavo 

it is not too 
I)laco (if tlio 
ng adcloj to 
•^ thdiisaiul) 
iivo 14 Imii- 
il.s, and Icavo 
3tly 14 Jmn- 
da' ])Iai!o of 

7 will not 
Jnes 7 are 7 
Ji'in;; tho 5 
units of tho 
tiniOH, loav- 
in the tens' 
urea fiii'tJior 
ni nation of 
phor. Tho 
!»o required 

-=9208. 
28, divided 



■ after tho 
>i" wo may 



' Hio divi- 
into thoui 
itlis times 
t <> in the 
onths ro- 
(i will go 
e 2 lum- 



DIVISION. 



83 



denominations of f]/c qu.)tient; wo m- v^ 1 o ■ ""'r''^'« 
put dowa m the quotient us nutn^ S^. s w ^.'^.'7^ 
liual remainder so small, that it mtiy be m'glecTid. ' 

75. Example 3.— Divide 473G5 by 12 
12)47305 ' 

* 3U47-08, &o. 
In thia oxampie, tho one unit loft Caftcr obtaininn- fKn 7 • 

ntr'i-^ tr ^'r. -r^-^i as io to^rsrit 
StZths'';;;;x e^^^^^^ "^^^^'"« to bo set down ; 

ine renins place ol tho quotient— except a CYphcr to kr.pr. 
tho following digits in their proper places 'Fm in ? !i ^ 
are by consequJnco to bo coLiXro. ^^^ 
12 will go int^ 100 l^undr^dZ 8 idr tl '^^ 
toliSuYtt iSn^^ ''' '-' -^« m whTn^rdosiro ' 
Example.— Divide 8 by 5. 

«-^5 = l'Sorl-37, &o. 
76 When tho pupil fully understands tho real deno 
n^inations of the dividend and quotient, he may proceed 
for example, with the following "^ pioceeu, 

5)40325 

In this manner :— 5 will not go into 4 5 I'nfn da a r 
find 1 over Cthe 40 bpi'nn. r>f Jv i • . ^"' ^ *i»ios 

produced it) 5Tnto 13%w- ^"' the den<imination which 

times and 2!,ver' 5 in l oJTtir:^ ^ r^' ^ ?"^° ^2, 6 

' ""^^^ ^"^ ^^^ remainder. 

-ligil. of the quotL^aro a/eo Str'lV' '' Ti"""" »'' «>« 

.ub.™t.. rro„ t.4^"^nf istsi;™ :i,'ir.ii' riL° 



84 



nivisiiotv. 



Thni, }f T) gooji 



tlio quotient (544.k5l7 w« '?„ ^,/^' ';• '", '""'"'f?' ["•' «xa»iplo, 

cnvi.Iond Huitc 1 to M, roCs"'? , '• "• '^'^'''••••^»T. to render tl.o 

>vlule, at tI.o «amo t me/wo e vo i s v^h'"' '''"■'•'"•• i'''^ '"'"•'"• 
coruort • *^" ^*-''^^ "** value unohangod; It bo- 



Thousands. Hnr„lreJ«. Tuns ' ir •, 

E«wh j.art being divided hv 7 f l,n hr ^'^' (^^^^f^^)' 
dividend, with tifeir ^iotiv'o q'uotonln^'^Jirbr''^^'^^ "' ^'^^^ 

and the nuestion is Zf./v *: '^ quotient in a lower; 

dividend~it3di?Lint de onf/n'V '" i''''''' ^''^ «" '"to tl'O 
venicnt way \Vo can n T^ T ^"'"» taken in a,;y con- 

Hhall have t^o add L t e LerdLonX.';;'- "^ '' 'I'' '''^' ^^° 
With the higher. '"^ ^''^^^^ denominations, unlesa we begin 

th^x^n5;Lf"r;f S divide:;!; 'Y r''^?' - p^* "-^o- 

it belong,, to that demmdnatiZ "!;';'''' ^''''^'''''\ '^' ^««'^»«« 
of time "(indicated by S of'thn. expresses wliat rumber 
tan bo taken from the coSInn'^"'^ "lenonunation) the <livi.sop 
thus tlie tons of the onot^^n?^ "" T"'^ ""^ *''« 'Jividend:- 
the divisor can be t ken fro^lT'''^; liow many tens of times 
hnndreds of ti^e mio ient tZ "" *,'"' ?^ *''« dividend; tho 
6e taken from tlie TuXds &c '"^ '''''^'''^' '^ '''^'' ''^ '^^ 
m^:ih::;it^;;j;-^i;So7^;,- l^f belong, to the total re- 
lower denomination t w i tti ' m •* "'i-^''" considered as of a 
He Asorv OK V -Ve a Jo f o 1 ''l"""- ^"'':^"^*^^ ^" *''« quotient, 
the highest deLm nation canablo n^ ■^'' ^"''"^'""^^^^ ^« «^ 
tliougli it may not cont^i n E J?„?^ ^^^"^^ i^ quotient; and 
press«. by a di<r it of one don., • I-'"'" 5' '''''^^^'' ''^ times ex- 
Lnbc., ot^im J^/p^oTed'ryTn^Ua^ «^"^'- ^^ -- 

t2!J:reSnXrtheS[t^5eS^S'"^ oach.produot, is the 
*) mucjj of it as is neoessirv f^ '/"'^ ^^ ^^''"3 down" only 
iooking for a d L t ?n ?£ JmnX';;' Pr''"^°^J^°*- '-Th^^' in 
^m not be ncc^essary tota^e Sy;^^^^^ quotient, it 

Of the dividend ; since hev canm f « n .""?, *^'' *'"''' °^ ""^ts 
<lre.s of times the aivi.iJ?„^;rfaS f^oI^thrriS.'^'^* 



DIVISION. 



85 



Acyvher mnst bo n.Mc.l [Sec. I. 28], when It ia rcquiml 

tlio orwe. except it oinoM botwcou thoiu un.l me .leciuml p-int. 
Hr:AsoN <.K VI.-Wo .....y continuo ti.o process of division. 
If wo plouso, a.s o„ff as it ih pusible to obtain .,uotio»ts of ani 
icnon.m.uoa. Q.iotients will l.o po<luce.l although th ore are 
nu lung.r any «.gn.nount figures L the dividend, to which wo 
can add tho succcHHivo roniaindora. 

78. Thfi BiiialW' the divisor the larger tho (motiont- 
lor, the smaller the parts of a given quantity, tho groat-r 
tlum- nuuiber will bo ; bt.t is tho least po.ssiblo tlivi- 
sor, and therefore any quantity divided ))y will <rivc tho 
larg(.st possible (^uoticnt-which in infinity. "Hence 
though atiy quantity multiplied ' by is equal to 0, any 
number divided by is e.pia! to an infinite number. 

It appears strange, but yet it is true, that-=:- ; for 

each is equal to tho gre.nlr.st pns.siblo number, and one, 
thcrefc 3, cannot be greater than another-the appa- 
rent contradiction arises from our being unable to form 
a true conception of an infinll.c (luantity. It is neces.^ary 

n.-m^ •^•'Tf '^'*\f'^"^ ^ ''' ^^''' ^'^«^^ "^dicates i 
quj tKy^inhnitely small, rather than absolutely nothin^r 

7J lo 2>roi-c Dlcis}^n.~lSlnmiy\y the quotient fv 

the divisor ; the product should bo equal to the divi' 

Uend, minus tho remainder, if there is one 

I'or, tlio dividcn.I, exclusive of t!ic ronaiuder, contains tho 

Ir'^t' rdivl.'": '• ""rr '""''r'''' '^^ ^^c quotient ";?.«! 

eoml nf r ',''i^''^'^'" ^'"'^ ""'"^'^'r of times, a quantity 
U i 1. tvH tirirnTr • "",'"' '''« r-naindcr, will be p^Uucelf 
it luiiows, tiiat adding tho remainder to tlie product of the 
divisor and quotient ahould ^ve the dividend. ^ ^ 



EXAMTLK 1.- 

T708 



n i.t i. 'L'^^32 ,_ 

rrovo that -— ^-=1708 
4 



PuooF. 1708, quotient. 
4, diviKur. 



sov and quotient, equal to the^dividcrlf'^' ^^'"'^"'^ "^' '-^'^■" 

ExARipi.E 2.-.l>r<,vo tliat ~"^^^- ioo-m ^ 

Proof. ' , ' 

rnooF. 
or 122^4 



122^1 
7 



**^^*^^=='iivMcu>l 



"limn:., tliediuniii V- 



7 

i: 2 




86 



DIVISION. 



2)78345 



EXERCISES. 



. <2) 
8)91234 



(3) 
3)67869 



(4) 
9)71234 



4)96707 



6)970763 



(6) 
10)134667 



12)876967 



(7) 
6)767456 



7)891023 



(8) 
11)37087 



(12) 
9)763457 



SO. iVhe>i the dividend, divisor nr hnth 
w hen the dividend contains cyphers— 

68 7 times, it will bo -ntrt fiRrin / -"^P^^^' if 8 w^l go into 
than 66) 100 tiSS ^orftLtTtiL^s^ir^oV.TJr-^ ^•'^^^*- 
i^AMPLE l.-What; is the quotient of 568000^4 i 

4 --14 J, therefore — —-. = 142000. 
E^mPLE 2.-What IB the q, ti^nt of 40G0000-.5 ? 
*; -SI 2, therefore —y— =812000 [Sec. L 39.]. 
81. When the divisor contains cyphers— 

can be taken fr- m it 100 tfmes^le^rSen. ''' ^^'^ '""^' ^ 

ExAMPi.E.-.Wliat is the quotient of -^ I 

ro °00 

- - 58 

■g=/ ; therefore ~-= 07. 



800" 



DIVISION. 



87 



(4) 
171234 



(B) 
37087 



12) 
33457 



■ contairi 
re apjtli, 
them. 

removo 
2rc liavo 

) be tlie 
e Uiviaor 
' go into 
d greater 



; 2 
39.]. 

we the 
cyphers 

times it 
'; 6 can 
times 6 



82. If both dindenrl and divisor contain cyphers— 
RuLE.-DA'id^ as if there were none, and move the 
quotient a number of places equal to 'the dfiL'enoe 

t^Hr'''-?^""^?'' '^- ^^P^^^*^ "^ *he two given quan! 
t ties r-if the cyphers in the dividend excefd thole hi 
the dmsor, move to the left; if the cyphers i^ the 
divisor exceed those in the dividend, move^to Te T^ght 



ExAMPLEa. 



(i) 
7)63 

9 



(2) 
7)6300 



(3) 
70)63 
" 0-9 



70)6300 700)630 



(6) 
700)6300 



eoo 0-9 —90 —0^ 9 

of Pvnit^l^- ^^^"WK the difference between the numbers 

83. If there are decimals in the dividend— 
KuLE.— Divide as if there were none, and move the 
quotient so many places to the right as there are deoi- 

The smaller the dividend, the less the quotient. 

ii^XAMPLE.— What is the quotient of -048-5-8 ^ 

48 -048 

g-— 0, therefore -^==-006. 

84. If there are decimals in the divisor— 
KuLE.— Divide as if there were none, and move the 
quotient so many places to the left as there are deci! 

The smaller the divisor, the greater the.quotient. 

JiXAMPLE.—What is the quotient of 54-i--006 ' 
54 54 ' ■ 

g-=9, thererore;^==9000. 

visor-"^^ *^'''^ """^ '^''*'"'^^' '"^ ' '*^* ^^^^^^^^ «n*^ <Ji- 

Kui.E.— Divide as if there were none, and move the 

quotient a number of places equal to the difference 



M 



68 



DIVISION. 



between the numbers of deeiraab in the two given quan- 
t.t.e« :_.f the deciraab in the dividend exeeed thor?n 
the divisor, move to the right : if the decimals in tim 
divisor exceed those in the dividend, move to the left 



Examples. 



(1) 

6)45 

9 



(2) 

6) '45 

•09 



(3) 

•05)45 

900 



(4) 

•5) -045 
•09 



(5) 
•005J_450 
90000 



(6) 
•05) -45 
9-00 



■ *],f^^''^;'~~?''^'^^ ^^ ^ *^^^« ^«r« i^either, and move 
the quotient a number of places to the left, equdlo 
the number of both cyphers and decimals. ^ 

Example.— What is the quotient of 270-f--03 ; 
-^=9, therefore, 270^-03=9000. 

in ?he d^koTl'" ^'""^'^' " '^' ^^^^^^^°^' -^ «^^-« 
Rule -Divide as if there were neither, and move 
the quotient a number of places to the ri4t eauS to 
the number of both cyphers and decimals. " ^ 

^Zi^^Z^S^^'"'^' '^' «- «^P^-- - the 
E.YAMPLE.~Whaf ia the quotient of -18^20 ? 

-^ = 9, therefo-e ■l| = 009. 
20 

The rules which relate to the management of cyphers 
and decimals, m multiplication and iS division-/hou4 
iumerous--will be very easily remembered, if the pupil 

. " ■' — """^ -^d^fcc. tu yu ine cuesi v) ciihor 



;iven quan- 
id those in 
nals in thu 
the left. 

I move the 
ivisor move 
ogether, the 
eir separate 



(6) 
'05) -45 
9-00 

and deci- 

md move 
equal to 

a la in the 



Icyph 



ers 



ad move 
equal to 

ra in the 



cyphers 
-though 
he pupil 
iihtir 



(13) 
8)10000 



(14) 
11)10000 



DIVISION. 
EXERCISKS. 

(15) 
3)70170 



(16) 
6)68630 



89 



(17) 
20)36623 



(18) 
3000)47865 



(1^) 
40)56020 



(20) 
80)75686 



(21) 
12)63-076 



(22) 
10) -08766 



(23) 
•07)64268 



(24) 
•09)57-368 



(25) 
•0005)60300 



(26) 
700) -03576 



(27) 
•008)57-362 



(28) 
400)63700 



(29) 
110)97-634 



88. When the divisor exceeds 12 . 

The process used is called Ions; division • thif i<? wa 
perform the multiplications, subtlaclLrr&c , nJl 
and not, as before, merely in the mind. ' This will be 
imderstood better, by applying the method of longdivi! 

Tat r ihanTr^ '• ^" ^^^'^^^~'^-- divisor noticing 
gi ater tlian 12 — it is unnecessary. ° 



Short Division : 
8)6763472 
720134 


the same by 


Long Division. 

8)6763472(7204:]4 
56 

16 
16 


> 




34 
32 

27 
24 

~82 

32 



dil?renM,aTof X'f 'r^"^^;^!-^ *^« ^^ ^'J the 
uuiuent paits ot tJie quotient, and in eacli case ,ei \\mo,x 









90 



DIVISION 




the rroducf mLlmct it from «,« corresponding portion of 
the <1 vKlond, write tlie romaindor, and irinn ffJnth^^, 
quu-od digu, ,>f the dividend, Ali this 3 beTne when" 

dent' .StH^-i^te™'- *" '^'^ '''' "' ""^ *"" 



III. Fidd the smallest number of dibits at ih<^ 7pff 

and set down, underneath, the remainder, if there is 
any. Ihe digit by which we have multiplied the divisor 
IS to be placed in the quotient. ^ 

. T* J? *.^''' remainder just mentioned add, or, as it is 
said " bring down" so many of the next' digit or 
cyphers as the case may be) of the dividend, as are 
required to make a quantity not less than the divisor 
and for every digit or cypher of the dividend thus 
brought down ^c.^^ .,^, add a cypher after the digit 
last placed m the quotient. ^ 

VI. Find out, and set down in the Quotient iht> 

nnmUr of times the divisor is contained if^hTs qUn! 

tity ; and then subtract from the latter the product of 

tlie divisor and the digit of the quotient just set down. 

J^roceod A^.th the resulting remainder, and with all that 

succeed, as with the last. 

A- ^?'}l *^®/^ '^ "^ remainder, after the units of the 
dividend have been " brought down" and divided, either 
place It into the quotient with the divisor under it, and 
a separating Ime between them [70] ; or, putting the 
decimal point in the quotient-and adding to the re- 
mainder as many cyphers as will make it at least equal 
to the divisor, and to the quotient as many cyphers 
mi7ius 071^ as there have been cyphers added to the 
remamder— proceed with the division. 



DIVISION. 



portion of 
^wi the re- 
done when 
lid be too 



the divi- 
e for the 



■ the left 
quantity 

lem, the 
contain ; 
there is 
e divisor 

', as it is 
igita (or 
, as are 
divisor ; 
Qd thus 
he digit 

mt, the 
s quan- 
duct of 
t down, 
all that 

of the 
, either 
it, and 
ing the 
the re- 
t equal 
lyphers 
to the 



t)0. ExAMPLK 1.— Divide 78325826 by 82. 



91 



82)78325826(955193 
738 

452 
410 

425 

410 

158 
82 

762 
738 

246 
246 

fnt?7« V '' -^1 ''l*^ ^ \ ^°' i"*^ ' ^ 5 ^»<^ it ^^i" go 9 times 

tV T^ ''i'',^^ P"* i" *^^e quotient. 
xvni L T-''^*,^*' ^^Shor denominations in the quotient 

if the DunTl ns hT '''' 'T'-'"^'' ^'^ P^-'^P«^ t« a«certo,in, 

ff titrtr!;h7ch th^e^roit" ^^^^^^^^^^^ '^'^'^ *^- -^--^ 

6 thnes'"! t'"^^* ^.'^'"'..^^ ^^^« 4^- i^^to^vhich 83 ^e 
LiiB uivihoi tiom 4&J, which leaves 42 as remainrlpr 49 
with 5, the next digit of the dividend, makes Sr^nowhith 

um me quotient. Ihe last remainder, 15, with 8 tliP n^vf 

digit of the dividend, makes 158, into whcr82 toi once^ 
leaving 76 as remainder:—! is to be nntln tK^ g?es once, 

rlToK^l ^ ^""Stit down expressed mut* 






t , ij 



Therefore 



78325826 

82" 



=955193. 



02 



DIVISION. 



Sample 2.-I)ivido G4212S4 by G42 

042)(;42i284('T^( 02 
042 ^ 



1284 
12S4 

ing tr'thrncx?d?'^' r??, ^^^r^"^ ----^-. nrl,.. 

1. TJh) next di^ of 't e 1 !n ' T-^'" ''' '^t'''^"^' ••^''^«^- *!'<^ 
«o digit in tJic^ not on h vhS "' '''' '^^'"'^ ^^''^^' g'^^'^ 
another cypher: and for sin l- r vf ' '^""^'^^l^^^tlj^ ^vo put 
down the next •X/ihn?''v ''."'' '*'"'^'^"- i» hi-bJivr 

gives^io romainderUve put 2 in'fT "'' f^'^''^"^' ''''''' ^"^ 
91 Whm-. +1.nv^ • "^P"*^ J" <^^"c quotient. 

division, adding deoimaS SesfoTl '^''^ T'^^ ^""*^""« "'« 
o uLcimai places to the quotient, as follows— 

ExAMPM 3.— Divide 79G347 by 847. 

847)79G347((i4010,'&c. 



3404 
3388 




convenient to have two on nSS Zi ■ '""^^ *''* ^'^ '' ^"^^'^ 



!i;fl 



DIVISION. 93 

6425x 54 

^± ViTsT&c. 

102 
64^ 

485 
432 

53, &c 
Rkason of n.-This, also, is only a matter of convenience 
Rkason OF III.-A smaller part of the dividen-l woul "ivo 
no digit m the quotient, and a larger would give more than 

Keason of IV.-Since the numbers to be multiplied, and 
the products to be subtracted, arc considerable, it s not so 
convenient as m short division, to perform themultiplicXns 
and subtrac ions mentally. The i^ile directs us to set c bwu 

onlTLitfe" "" ^"^^■^^^^' '^^^"«« ''' ^^^*- - *^- -- 

liFAsoN OF V.-One digit of the dividend brought down 
would make the quantity to be divided one denomination lowe? 
than the preceding, and the resulting digit of the ouEt 
also one denomination lower. But if we are obliged tTbrfn - 
down two digits, the quantity to be divided is fwo denoiS? 
nations lower and consequently the resulting digit of tlie ot^o 
tient is ^^/»fl_ denominations lower than the preccdino-i-which 

uro-^a'c^vnliet^^'/.r*'*"" ^'^" '■ 281 is exp^ess'^St; 
using a cypher In the same way, bringing down three 
S't' '/ the dj^j^^^^^ ^^^j^^^^ ^^^ denomination thrle places 

lower^£n the hZ^ '' "f ''''''''''' ""'^^^ clenominSs 
lower tnan the last— two cyphers must then be used Thn 

T.^TT'^'V'''^^? ^"'^ an/number of characteis whether 
significant or otherwise, brou-ht down to any remainder 

J^^l^fl ""^ ^^.-^^« ^"bt^-"'^* the products of the different 
parts of the quotient and the divisor (those different Ss of 

are ?ounS th^„'/°^ ^^^ '^7'' «"«««««-ely accorSg^aB they 
are tound), that we may discover what the remainder is from 

w W J'A'''' *', "^Pf '* *^^ "^^* P^''*^*^" «f the quotient FroS 

no 3ecTmaYs?nT1? --'^ ^IP' '' ''■''''''''' *'^<^*' if there ar^ 
noaecimaism the divisor, the quotient fifmre will alwiv^ hi 

It is proper to put a dot over each dimt of tlie divi- 
dend, as we bring it down ; this will prevent our Ltei- 
tmg any one, or bringing it down twice. ^ 

94. When there are cyphers, decimals, or both, tho 



94 



DlVIiJJOX. 



9o. lo prove the Ihmion.—mxMMy the ouoticnt 
by the divisor ; tLe product .should be iL\ to the div - 
dcnd, mums the remainder, if there is anV '791 
^^.^io^Fovo It by tho method of "casting out tho 

KuLE.— Ciist tho nines out of the divisor, and tho 
quotient ; multiply the remai.ider.s, and cast tlirni it 
fiH„u tiieir product :-that which L now left oi^ to 

out of tlie dividend minus the remainder obkined from 
the process of division. «"ucu uom 

Example.— Prove that ~J!~= 1 181 .3 

Con^-ider^daBa'^ 
;, ,. ^ '^'*-^^776-2 = 037/4. To try if this bo true, 
Casting the nines from 1181, the remainder is 2. ), . 
.. ". " i^'Oin 54, „ jn 2x0 = 

tasting the nines from G3774, tho remainder is . .0 

The two remainders are equal, both beino-Q- henco tl.n 
multiplication is to be presumed right, ancf «Zi Iv 
the process of division which suppos "s it. ^^'"'^^'Ititntly. 

The division involves an example of multiDlicatinn • sJn^.* 
the product of tlie divisor and q,loticnt oi^ghfto be cnuaMo 
the dividend minus the re.uainder [7'J]. lieLe in mot^nc? 

!idiri^^;^zirir^' - ^'^y e.pij.;rc£;nf 



EXKRCISiCS. 



(30) 
24)7054 



(31) 
15)0783 



318f3 



(32) 
10)5074 



452^, 



(33) 

.* 17)4075 



35410 



275 



(34) 

18)7831 



(35) 
10)5977 



435A 



(30) 

21)G78r 



(38) 
23)707500 



3141^ 



(39) 

390)5807 



(37) 
22)9707 



323 



443|i 



33309^5 



(40) , 
1400)0707000 



14-8897 



4035-3425 



ic quotient 
to the divi- 

g out; tho 

•, and tho 
tlip nines 

1 ought to 
tho nines 

iued fioni 



3-becomos 
truo, 

2x0 = 

. 

henco the 
iso(;[uontly, 

On ; since 

i equal to 

a proving 

[54], we 



(33) 
'17)4075 

"275 

(37) 
2)9707 

'ml 

000 . 
335-3425 



4 



(41) 

250)77670700 



303424 •00y4 

(44) 
64-26 )123 •705 86 

2 -2803" 



DIVISION. 

(42) 
67-1^^42 
•002 



96 

(48) 
•163 ) -8297 49 
5-4232 ' 



(45) 
14 -86 )269 -0625 
18-75 



(46) 
•0087 ) 655 

150000 



In example 40— and some of those which follow— after 
obtaming as many decimal places in the quS as I'o 
deemed necessary it wiU be more accurate to cons der t^ic 

haTf'o?ftT Z?"u *^f ^^\r^r («i^«e it is more than one 
halt of It), and add unity to the last digit of the quotient. 

CONTRACTIONS IN DIVISION. 

96. We raay abbreviato the process of division when 
rflf f^%«^ajy decimals, by cutting off a digit to the 
right hand of the divisor, at each new di4 of the 
quotient; remembering to carry what would have been 
«n tTff th^ the mdtiplication of the figure neglected!!! 
unity if this multiplication would have produced more 

25r&c' T59T ^^ ' ^ '^ ""'"' *^''' ^^' "' ^''' *^"" 

Example.— Divide 754-337385 by 61-347. 
fli oA?!n.TLT^'''^- Contracted M^od. 

61-347)75^|33 7385(12-296 61-347)754-337385(12-296 



C1347| 

14086:7 
12269 4 



1817 



33 



I226j94 

590J398 
652 123 



38 
36 



2755 
8082 



i|46730 



61347 

14086 
12269 

1817 
1227 

"590 
552 

li 

37 



DC 



I'lVlSiOX. 



;i'u-ntly. tho portions of the dividcna from vMch th ^ T'n 
li-ivo been Hubtracted. "What hIiouI.1 i.uvn h •^ .^ '"''^ 

the multiplication of ti.e digi nSteHin^^^^^^ '■"'" 

97. Wo may avaU oui-selves, in diviaion, of oonfrJ 
vances very similar to those 'used iu m;Uil>lS;,n 

To divide by a composite number— 
lluLE— Divide successively by its factors. 

Example. -Divide 98 by 49. 49=7x7 

7)98 
7)14 

"2=98--7x7,or49. 




98 If the divisor is not a composite number wp 
canno,as in multiplication, abbreLte' tC process 
oxcept It IS a quantity which is but little less^than a 

c^edTthe """^ f"^' *^" ^"^*^^"' - the _d 
Tn.^- A- ?''''"' i'^'"^"' «^^ ^i^'i^e the sum by the 
precedmg divisor. Proceed thus, adding to thTrcLi ' 
der in each case so many times 'the foregoing .uoHcn 
or i s'nffiT^f '^'^^^^ '^' Siven divisor untilV "xlc^, 

i?^. atcTlLT/l^P- ''""''^*^'^\*^ *^^ exact quotien 
IS obtained--the /c5^ divisor must be the given, and not 

the assumed one. The last remainder will be the ti ue 



tilVlHlON. 



fff. 



K.VAM,.r,K.~I)ivido l)87(5r.;342r, by 998 
9870G;j.,425=t)S70(i342r)^10()u" 
i97r>.jr.l=ys7lTo33^2qp425^U)00 
4.JU1=.IU752^75T4-1000. ■' 
0-7..0U0=4x2-|:701>]00() 
. 0()l4)4()=-7x2.flT^1000 
OU00..420=:.ur^2:p4_^100(). 
00004,,0208=aT^2+^^908 
icnutiS^" '"' ^^^^^««* - «-^004, ^a -0208 is the lu«t 

r 987G03 
1975 

all tlie quotients are • '-^ » 

07 
O-Ol 
I 0-0004 V ' 

The true quotient is 9890427lO4 », ^''•'^'** 

And the true ren.uindor 0.0908 ' Z u 7" "'' '^' .l"""""''- 
Uj 1 ^ u i'-u«, or tlie iast reniaindew 

or the part qf it ju,t UiSd lin \ ^»-«'n t^'e dividcul, 

tlio third lino 4701 ,nV J^''\ ^I"otient. Thus in 

as quotientrand ^Tof ^itf s'jl'rt' ^ ]T^ ^ ^^^^^ 
701 us ren/ainde/ 4-7 wo ] ^'^^'^^^-that is, 

to the Icl/of tie decinn \^/' ;>^^^P.Vin^ four places, al 
pv- units as quj^;^ r^'al^^tlf^^ ^ ^^««' 
liue), one is a decimal i.!.. *! -^ ^ ^^" *^^ »ext 

tenths; and in OlO-JO InJt y \ ■ ^' "' ""= '"''ier 

four piaoos are doo m- iT f j '^'r'""" '™ "»' "^ 'he 
dreJths, &o. ""-'"»■"''. "« quotient must be huu- 

plied, and\he .:,,'';:;".";' '?"»/'- V-tient multi^ 



o-ddcd, be] 



onix 

o 



•oiiiaindcr to wliich the product 



t.ts 






IS to be 



98 



DIVISION. 




^ 





47 
4H, 
4!), 
60, 
CI. 
62. 

ca. 
ct, 

Go. 
(SQ. 
C7. 
68. 
61). 
00. 
v(il. 
(52. 
(53. 
(54. 
G5. 

m. 

G7. 

08. 



EXKHCtNKa. 

, r)0789-f.74l==70*7;« 

47H!M>7-M)71=4l>;;JL'f. 

1)77070 +'17(50(3=20 ?hZI. 
607807 -^8 12=074 •"!• 

78(57074 -f-071 2=816 W. 

JK)7O7()O-f-457O()O=(J-7l03. 

07051 68 -^7894=8r)7. 

tt7470-^.;{l)00=17-3. 

0OO0O-f-47()0O=l -4490. 

70707 -^ 40700=1 -8802. 

(51M692-T-704;^24=8. 

9070744-^9] 0070=1 • 0.'J29. 
740070000 -^741000=998 -7449. 
94 10(507 1 1 1 -f-45078=200043 • 1 1 32. 
45407(5000-^400100=1 1 34-9003. 
737(547(57(57 H-a46(570=:2i::;39 -049. 
47 ;6782975-^20• 175=1 -8177. 
47 -(355-^4 -5=10 -59. 
760-98-^7G-7301 2=9 • 800. 
75 -3470+8829=190 -7798. 
0'l+7-0345=0-0000131. 
5878+0-00090=5002083-33, &a 



6J. If £7i)00 were to be divided between 5 persons. 
how nmch ought each person to receive > A71S. it; 1 500. 

70. Divide 7560 acres of land between 15 persons. 

_,,,.., ^ , ^^?^. Each will have 504 acres. 

71. Dmde £2880 between 60 persona. 

„r. rrr^ . , ^«-'- ^acU will reccivG £48. 

72. >Vhat 13 the ninth of £972 ? Am. £108. 

73. AVhat is each man's part if £972 be divided 
among 108 men ? Ans. £9. 

74. Divide a legacy of £8520 between 294 persons. 
_. -^. „ -^ns. Each will have £29. 
7o. Divide 340480 ounces of bread between 1792 

persons Ans. Each person's share will be 190 ounces. 

/6. Ihere are said to be seven bells at Pekin, each 
ot which weighs 120,000 pounds ; if thev were melted 
up, how many such as great Tom of Lincoln, wcighinff 
9894 pounds, or as the great bell of St. Paul's, in 
London, weighing 8400 pounds, could be made from 
them ? Ans. 84 like great Torn of Lincoln, with 8904 
pounds left ; and lOOlilco the great bell of St. Paul's. 

77, Mexico produced from the year 1790 to 1830 a 



I>IVIS10N. 



09 



ive £48. 

divided 

persons. 
Lvc £29. 
m 1792 

ounces. 
:in, eacli 
! melted 
vcighiug 
lul's, in 
ie from 
th 8904 
Paul's. 

1830 a 



f 




I" 5«7,01y,7-IO iuil,.3 i, ,' ",t ',- ' ""P"™ "l"'"' 



ifl29'0775 in 18'57 28b 13o6 m 1740; and 

I.alf of ,„en ?i I^ow , ?'" '"'• T ^ '"''"'"■'^ ■""• » 

* eo^:"ro:'L^;,t!:rf„,S:!i ^^ ^-^-'^ '^ 

I'-XAJii.LK.-Divldo 84380848 by 87532 
87532)84380848(964 



5G0204 



350128 

' the dividend to 

vnlaindor after s,h..u.ttitl,o""'''.'l •*''^, *'^^^ ^«) ' 2 (tiio 

'"^ f'.'H-ried from 98 an.] 1 fT^^" ^? *!,'''°'"' '^ + '^''e 2 to 

rowed ulu.n „/ ' ^1 \ *" «<»"ir)en,siito for wl^.A ,„„ i.... 
lowta When wo coiwideved in ti 






e duidund as 10) ; (ti: 



19 



100 



DIVISION. 



romalnder when we subtract tlie risht liand digit of 48 from 

11 om the 48 )j (tho' remaindcu- after eubtractiii;]; the rio-hf 
hand digit ot 67 from 3, or rather 13 iu the dividend), "id 
{J times a + the G to bo carried from -the 07 + thel for 
what wo borrowed to make 3 in the dividend become 13) : 
dendT ""^"^^ '''' ^^^^'^ subtracting 79 fro^n 84 in the divi' 

j.n;t!,nf'f .^f *' '° .*¥ ^^^^".theses are merely explanatory, 
and not to be repeated, the whole process would be 

Jmstpart, 4, 18 ; o. 28 ; 2. 48 ; 07 ; G. 79 • 5 
Second part, 8. 12; 2. 19 ; 1 32; 0. 45:5 53-3 
lhirdparfc,8;0. 12; 0. 21 ; 0. iO;0. ^5 ; ' 
J he remainders m this case boincr cyphers, are omitted. 
All this will be very easy to the pupil who has prac 
tiscd what has been recommended [13, 23, and 651. 
I no chief exercise of the memory will consist in recol- 
lecting to add to the products of the diifcrent parts of the 
divisor by the digit of the quotient under consideration, 
what IS to be carried from the preceding product, and 
I mty bes.des-when the preceding digit of the dividend 

hnnd d?>"''r!u-'^ ^^ ^2 5 then to subtract the right 
hand di^it of this sum from the proper digit of the 
dividend (increased by 10 if necessary) 

QUESTIONS FOR THE PUPIL. 

1 . What is division ? [66] 

3. What is the sign of division ? [68] 
4 How are quantities under the vinculum, or united 
^^ TAr'°° ""^ multiplication, divided ? [69] 

12 nn, r* r *V '?^? ""^'^ *^" ^^'"'' do«« not exceed 
12 nor the dividend 12 times the divisor.? [70] 

piit Xn .f ' r'^ ^^^. ^^'' ''''''''' ^f '^' different 
paits, when the divisor does not exceed 12, but the 

divi.lend IS more than 12 times the divisor ? [72 and 771 
7. How is division proved ? [79 and 95]. -•' 

b V\ liat are the rules wlion the dividend, divisor, or 

both contain cyphers or decimals.? [80]. 

9. AV^hat is tho rulo •ii)<l wl--^ ^i--. -^k-. c •. 

,.^ ";- ^""-7 •">5 wJiac are the reasons of 13 

different parts, when the divisor exceeds 12 .? [89 and 93j . 



GREATEST COMMON MEASURE. 



101 



10. What is to be done with the remainder > r72 
and 89j. ' '-''* 

11. How is division proved by casting out the nines ? 

[95]. 

12. How may division be abbreviated, when there are 

^lecimals i' [96] . ' 

33. How is division performed, when the divisor is 
a composite number ? [97] . 

• v^; v!T ,'^ *^^ division performed, when the divisor 
i.s but little less than a number which may be expressed 
by unity and cyphers ? [98] . 

15. Exemplify a very brief mode of performin<r divi- 
sion. [99]. / ° 

THE GREATEST COMMOxY MEASURE OF NUMBERS 

100.^ To find the greatest common measure of two 
quantities — 

IluLE.— Divide the larger by the smaller ; then 
the divisor by the remainder ; next the preceding 
divisor by the new remainder :-continuo this process 
until nothing remams, and the last divisor will be the 
greatest common measure. If this be unity, the ffiven 
numbers arQ prime to eack other. 

and 4248''' ~^'"*^ *^® greatest common measure of 3252 

'"va ?)4248(1 
3252 



99G)3252('3 
2988 



264)990(3 
792 



204)264(1 
204 



60)204(3 
180 

"24)6012 
48 



12)24(2 
24 



V im 






102 



GREATEST COMMOxV MEASURE. 



fh^i'.*^"'^^''^* remainder, becomes the second divisor 2G4 
the second remamder, becomes the third diW&o 19 
the last d.v.sor, is the required greatest cLnrS^easuro ^ 

tied that ' f any quantUv m^ZZ^ ^'' A^' ^^^^^ *° ^^^ «a««- 
any multiple of t^,2t Xr » thus ff T^n'-': %f^} ^^^^'^^^ 

sum. for if 6 g. into 24, 4 timet and^iS^ ? tL^U ;n/evt 
dently go into 24+36. 4+6 times :--that is, if ^=^, ^nd^^- 
6,-6-+?=4+6. • 6 6 

ence between the numbers of ti,^P«u V ^*^^ ""^ *^« ^i^er- 
due to this difference, l^stte^TtalLr!^ ft^^e ^^ 
of tmies ..-that is, 8ince^i=6, and ^=4, ?^_24 X 36441 

oth^rliSSCr ^'^^^^^ e^TeSIHorrect with any 

and that it is L ^2^ folo'n 1^^ ^ "^^"^^ ^ 

we fiXraH 2leTsrer24'^tf"^ ^* *^^ ^"^ «^ t^^« Process, 
a multiple of 24?^ thl'slTiSw?' ^'' ^f^^^^ ^* '^ 
each of them) or 60- and IftThlv 4^-^''^''^® '* measures 
and 180+24 Twe hnv^ S^ • . because it is a multiple of 60 • 
these) or ib4 an Jo^teo-'or 2rT^ that it measures each of 

pie of 264; and ?92+2ot or 996 ^'hS^QkI^' ^'''''.'' ^ ^^^l^" 
and 2988+264 or 3252 Innp J vl' ■ ^^^^' * multiple of 996 : 

996 or 4218 (throJlSi v'erLmL?)''^^.^,^^^)!"^ '''^+ 
each of the given numbl-s an<f^ fS ^^^^^^'^^^^ it measures 

some other be greater ir/n ?tl ^'.^'"^on measure. If not, let 

process) measu^r7nf4248 and s'S'TtTS 'T."' *^^ ''^ ^' '^' 
measures their difference 996 aniio««' k^' supposition), it 
of 996; and. because irmefur'es 3252 n'^^S*^"^ "^^^*^Pl<^ 
their difference, 284 • an.l 7Q9 tL ' '^'''^ ,^^^^» ^* measures 
the difference between j'6 and Tgr^'^of^^^^^^^^^ ^'^^ 

between 26 4 and 204 or GO • and 1 «0 h ' ""'^ *''^ difference . 
ami the difference between''-'04 and 1Ro"''"h *" '"^^*'P^« ^^ ««; 
a multiple of 24; and tre'd^ffl^l ^'^t^^A^ '^^^^^^^ 
But measunng 12. it cannot be greatc^,- tha^ 12 """ "° "^ ^' 



GREATIuST COJIMOxV MEASURE. 103 

rntJre\TZlZnZZi'f'T.' *^* '^"^ other common 
sequently that 12 s tH 'IS' n ^''' *^''^ ^^~'"^^ ''«"- 
rule might be in-ovea fiom r!nv !> '"°" measure. As the 
it is true in all Lses ^ °*^'''' ^'^^"^JP^^ ^^"^"y well. 

^ 104. We may here remark, that the measure of two 

, -my quantity, tiio diitit of whoso bwest donnmin„n„„ 
.3 au even .mn.bor i« aivimhh by 2 at ka t '"" 

Any number on, ,ng in 5 is divisiblo by 5 at least 
j^ Any nttmber onding i„ a cypher is div^isib?^ l'; io at 

EXERCISES. 

ancl-lSSV"'lt.f ''' ''"^"^"'^ "^^^'^"^^ •^f 464320 

2. Of 638296 and 33SS8 ? ^,w q 

3. Of 18996 and 29932? ^t 4 

4. Of 2G0424 and 54423 > Ans 9 

5. Of 143168 and 2064888 .P Ans'. 8. 
t). Of 1141874 and 19823208 > A^Lf. 2. 

ihlni^'Jl'Jl ^'''''''' — «« --sure of more 

of lh4 1 If ''^'"''"''^ "^"'^•''^^•^ a»d a third ; next 

ot this hist common measure and a fourth &e S 
last common measure found will v.. +i "'^"' ^'^^ J-^ie 
measure of all the given ^il^^^^ '^' ^''''''' ''^^'^ 

SOotTnd 673^^^'''^ '^'' ^''^'''^ «^°^^«° «^oasure of 679, 

.n-eatost common measure of 7and 6731 71?' *''^' '^' ■'''" 
numhor), for 6734 -^7-060 wIH.^? • ^^'"^ remaining 

7 is the required numbTr ' ""^ ^•'^•"'•^•igli'. ThereforS 

TSoi'^mfHil '"^ "'^ '''' -^^^^^^t ««^'"non ;^a«ure of 0:^0, 









104 



LEAST C0iv/1\I0N JMULTIPLE. 




The greatest common measure of 93G and 73G is 8 and 
tlic eonnnon measure of 8 and 142 is 2; therefore 2 is 2o 
groutesfc common measure of the given numberr 

fncf n toCfa'ft'^ .go through all of themin succession ; 
nro to l..S,r„T*i * '* '^ ^^^ greatest common measure, we 

used toind tU .T'""'"''''^'"""* '^ ^^^ fi^'«t process, or tlm? 
usta to hnd the common measure of the two first numbers 
tond proceed successively through all. numDers, 

EXERCISES. 

i'tLo^^^^jH^^ greatest common measure of 29472 
176832, and 1074. Arts. 6. -»^'^, 

8. Of 648485, 10810, 3672835, and 473580. Am 5 

9. Of 16264, 14816, 8600, 75288, and 8472. Ts 8. 



THE LEAST COMMON MULTIPLE CF NUMBERS. 

titils— ^"^ ^""^ ^^"^ ^'''''* '°'''"'°'' """"^^^P^^ °^ ^^'^ ^^^"- 

lluLE.— Divide their product by their greatest com- 
mon measure. Or ; divide one of them by their greate t 
common measure, and multiply the quotient by the 
o lier-the result of either method will be the required 
least common multiple. " 4""^"- 

KvAMPLE.-Find the least common multiple of 72 and 84, 
1-. is^their greatest common measure. 

1^ = 6, and G X 84 = 504, the number sought. 

108 Reason of the Rui.E.-It is evident that if we muU 

SiniT'^i "^""^'r^ *°Sether, their product wHl be a 

tiultiple of each by the other [30]. It will bo easv to find 

he smallest part of this product, which will stiKe their 

of^r'nniilMB- '-'71 r^"""'^^' '''" 1^^'^^' ^'^^^^ of t'^e factors 

Irotlot^nf^Xr f ^ ''"•X ""'"^^^' '^"'i multiplied by the 

& J -V 1 ,""'■, ^'"'^'''■'' '' "^'""^ to the product of all the 

.io-.« ^n-idcl by the same number, ironco "'>. .pd ri L; ! 



LEAST COMMON MULTIPLE. 



106 



$m 



2X84 



~^-^ (the nineteenth part of their producc)=I?x84, or 72 x 

J. y 



u 



_. Now if '-- and __ be equivalent to integers, ^x84 will be a 



multiple of 84, and°|x72, will be a multiple of 72 [29] ; 
andi--g_, L.X84, and 72 X~| will each be the common 
multiple of 72 and 84 [30]. But unless 19 is a common measure 
of 72 and 84, j-^ and _ cannot be both equivalent to integers. 

Tl.erefoie the quantity by which we divide the product of the 
gixen numbers, or one of them, before we multiply it by the 
ctlior to obtain a new, and less multiple of them must be he 
c.mmon measure of both. And tbe multiple we obtain w 11 
cvi.lent y, be the least, when the diviso ■ we select ^stlo 
grea es quantity we can use for tiie purpose-that is! e 
greatest common measure of the given numbers 

It follows, that the least common multiple of two 
numbers, prmie to each other, is their product. 

EXERCISES, 

1. Find the least common multiple of 7S and 93 
Ans. 2418. ^ 

2. Of 19 and 72. Ana. 1368. 

3. Of 464320 and 18945. Ans. 1759308480 

4. Of 638296 and 33888. Ans. 2703821856." 

5. Of 18996 and 29932. Ans. 142147068. 

6. Of 260424 and 54423. yl%5. 1574783928. 

109. To find the least common multiple of three or 
more numbers — 

KuLE.— Find the least common multiple of two of 
thcin ; then of this common multiple, and a third ; next 
ot this last common multiple and a fourth, &o The 
last common multiple found, will be the least common 
multiple sought. 

TA-AMPLE.-Find the least common multiple of 9, 3, and 27 
^ '^ IS the greatest common measure of 9 and 3 ; therefore 

g X 3, or 9 is the least common multiple of 9 and 3. 

'^^9 is the greatest common measure of 9 a«7 ; therefore 

^ X 9, or 27 is the required least common multiple. 



I, 
ii J; iii 



106 



LEAST COMMON MULTIPLE. 



that^^ili'lir/"'.'""*^ RuLE.-By the last rule it is evident 
tnat J7 IS tl e least common multiple of 9 and 27. But since 

a nuauri* %"•' 'IV'V, ^^'"/'^ -.^^ multiple of U, mu^t^airt: 

tlii is Imanif ."'"'i" ^'"''' ""'""^"^ "^"^*'P^«' ^e^a'^so none 
tnat IS smaller can be common, also, to both 9 and 27 since 
they were found to have 27 us their least common multiple 

EXERCISES. 

it 

aIs. mls'^'^ ^'''''^ ''''"'""'''' multiple of 18, 17, and 43. 

n Ri .^?' 3 ^^^ ^"^ ^1- '^>^'- 1265628. 

10 ^;^^-«?f ?'.rp,f ^' ^^^ ^^^2. .in.. 2937002688. 
10. Of 53/842, 1G81<J, 4367, and 2473. 

11 Of oir'jp o.iot. n... ^«5. 8881156168989038. 
li- Of 21636, 241816, 8669, 97528, and 1847 

Ans. 1528835550537452616. 

' QUESTIONS 

1. IIow is tho greatest common measure of two quan- 
tities found .? [louj. ^ 

2. What pvinciples are necessary to prove the correct- 
ness ot the rule ; and how is it proved ? [101, &c 1 

o. llow IS the greatest common measui-e of thi-ee, or 
more quantities found .? [105]. ' 

4. How is the rule proved to be correct ? [1061 

o liow do we find the least common multiple of two 
numbois that are composite .? [107]. 

6. Prove the rule to be correct [lOS]. 

7. How do we find the least common multiple of two 
prime numbers .? [108.] 

5. How is tho least common multiple of three or 
more numbers found.? [109]. 

9. Prove the ;ule to be correct [110]. 

^ In future it will be taken for granted that the puB^ 
IS to be asked the reasons for each rule, &c. 



t 



107 



SECTION III. 



19 1 

m 



quan- 



REDUCTION AND THE COMPOUND RULES. 

The pupil should now be made familiar with most of 
the tables given at the commencement of this treatise. 

"^ -REDUCTION. 

t 

1. Reduction enables us to change quantities from 
one '^nomination to another without altering their 
value. Taken in its more extended sense, we have often 
nractised it already : — thus we have changed units into 
tens, and tens into units, &c. ; but, considered as u 
separate rule, it is restricted to applicate numbers, and 
is not C0nfin|4 to a change from one denomination to 
the Tiexi higjier, or lower 

2. Reduction i» either descending, or ascending. It 
is reduction descending when the quantities are changed 
from a higher to a lower denomination ; and reduction 
ascending when from a lower to a higher. 

JRjcduction Descending, 

3. RuLE.^^Multiply the highest given denomination 
by that quantity which expresses the number of the 
next lower contained in one of its units ; and add to 
the product that" number of the next lower denomina- 
tion which is found in the quantity to be reduced. 

Proceed in the same way with the result ; and continue 
tlie process until the required denomination is obtained. 
Example. — Reduce £6 16s. OjcZ. to farthings. 
£> s. d. 
6 „ 16 „ Q\ 

136 shilling8=£6 „ 16. 
12 



1632 pence = £6 „ 16 „ 0. 
4 

C520 farthings OB jC6 „ 16 „ Q\. 



" * 1 



l a— l: :^'i 

" 11 ■! 



108 



REDUCTION. 




6 aro 24, and 1 nrA 9*^ 4 .. ^"^ ^ ^^°?6« ^ ^ o 12. 4 tunes 
4 SrS m'ny' fertile ."^"^"^'""'^.■'™™• pence to 

EXERCISES. 

93312^'''" '^''''^ ^^'^^""°' ^*" ^^^~S P^^^e-^ ^^«. 
2. How many shillings in i2348 > Ans. 6960 
^. How many pence in ^638 10^. ? Ans. 9240 

4. How many pence in ^58 13.. ? Ans. 14076 

5. How many farthings in £58 135. ? Ans. 56304 
67291 "'"""^ farthings in £59 13.. G^d. ? Am, 

7. How many pence in £63 0.. 9d. ? Ans. 15129. 
a!s\ 1864. ""'"^ P'""^^ ^" '' ^^*-> ^ ^^^-^ 16 ife- -^ 

.l.^: 1^68. '"''''^ ^'"''^' ^"^ ^'^ '^*-' ^ "i"'- 1^ J'^- -^ 
^^^5^, ?°^ "^^"y grains m 3 lb., 5 oz., 12 dwt- 16 



IlKDUCTION. 



109 



11. How many grains in 7 lb., U oz., irrdwfc./M 
grains ? Aiis. 45974. 

12. How many hours in 20 (common) years? Am 
175200. 

13. How many feet in 1 English mile ? Ans. 5280 

14. How many feet in 1 Irish mile > A7is. 6720 

15. How many gallons in 65 tuns .? Ans. 16380 

16. How many minutes in 46 years, 21 days, 8 hours, 
56 mmutos (not takmg leap years into account) ? Ans. 
*4208376. 

17. How many square yards in 74 square English 
perches ? Ans. 2238-5 (2238 and one half). 

IS. How many square inches in 97 square L'ish perch- 
es.? Ans. 6} 59H88. ^ 

19. How many square yards in 46 English acres, 3 
roods, 12 perches ? Ans. 226633. 

20. How many square acres in 767 square English 
miles ? Ans. 490880. ^ 

21 . How many cubic inches in 767 cubic feet ? Ans. 
I32o376. 

22. How many quarts in 767 pecks .? Ans. 6136 

23. How many pottles in 797 pecks ? A71S. 3188. 



Reduction Ascendviw. 



5. 



required denomination is 
. hings to pounda, &c. 



EuLE.— Divide the given quantity by that number 
ot its units which is required to make one of the next 
Higher denomination— the remainder, if any, will be of 
the denomination to bo reduced. Proceed in tlie same 
manner until thi ' ' 1 o ■ . , , 

obtained. 

Example.— Reduce 8(.,, 
4)856347 
12)214086f- 

892 ; „ 01=856347 farthings. 
4 divided into 85G347 farthings, gives 214086 ponce arul 
3 farthings 12 divided into 2!4ol6 pence, gives 17840 
'i«.lo°^' r^ ^ ir^V,9«- 20 divided into 1^840 shillings, a<ives 

7}7xr ''" i""" «^iiJ^ings; there ie, therefore, nothing in the 
Bhilhngs' plac . of the result. ** 



no 



HKDUCTION. 



Wy (livi.|r, by 20 if ^0 (llvi.lo l.y 10 lui.l 2 FSoc If 971 

any, [N,o. 34] w uch will then bo tbe unit« of sbilli, .^s 

11 t b., result ; and tie quotient will bo tens of shillings :^ 

i.viding tbe xa tor by 2 gives the pounds as quotion? and 

di;;:srlS^'^' '' ''^^^ ^-^ -^ - ^^- Quired q.«.. 

VoIIk-^"''"'"'" "*^ T"^ Rin.K.-It is evident that every 4 
Earthings are equivalent to one penny, and every 12 pence to 
«neslnll.ng,&c ; and that ^hat is left after taking away 4 
far b.nga as often as possible from the farthings? intTsfbu 
farthings, what remains after taking away 12 pence as often 
as possible from the pence, must be pence, &o ^ 

7. To prove i?€r/%c^207i.— Reduction asccndin'' and 
acscendiDg prove each otljer. ^ 



Reduction < 



f 



* s. 

20 

417 
12 



d. 



farthirgs. 
4)20025 



Reduction j 12)5006| 

20)417'„ 2 



5006 
4 

4)20025 



Proof . 



Proofs 



1 



12 )5006| 
2 0)417 „ 2 
^20 „ 17 „ 2] 



^20 „ 17„ 2« 
20 

417 
12 




20025 farthings. 



RXERCISES. 



nooto ^^"""^ "'''''^ P^'^^^ ^" ^3312 farthings ? Ans. 

25. How many pounds in 6960 shillings? Ans ^2348 

roon a7 ""^""^ P''''''^'' ^'^^ ^^ ^^^ halfpence ? Ans. 
pi/fi Us, od. 

JI\ ^""Z^^"^^ ^^^"^^^^ &c. in 7675 halfpence ? Ans. 
Jblo 19*. 9^a. 

28. How many ounces, and nounds in 4352 drams? 
Alls, 2/2 02., or iv ib. 



REDUCTION. 



Ill 



29 How many cwt., qis., and pouuda in 1864 pounds > 
Ans. l(i cwt.,2(ir,s., 1(5 lb. i -". 

30. How many hundreds, &c., iu 16G8 pound.s. A,ls 
14 cwt., 3 qns., i6 lb. ^ 

31. How many pounds Troy iu 115200 frvahiH > 
Ans. 20. " b • 

J ^^vIm'^^ '""""^ 1'°''°'^'* '" ^^'^''^20 ^^' avoirdupoise ? 
tills. d720. 

.,.2^1* ^?^^ J"*"^ lio<rsheads in 20C58 gallons ? Ans. 
127 hogsheads, 57 gallons. 

34. How many days in 87G0 hours ? Ans. 365 

35. How many Irish miles iu 1S34560 feet.? Ans. 

i lO. 

^^- ^low many English miles in 17297280 inches ? 

37 How many English miles, &c. in 4147 yards > 
Ans. 2 miles, 2 furlongs, 34 perches. 

iiS. How many Irish miles, &c. in 4247 yards ? Ans 
i mile, 7 furlongs, 6 perches, 5 yards. 

39. How many English ells iu 576 nails ? Ans 28 
«ils, 4 qrs. 

40. How many English acres, &c. in 6097 square 
yards ^ A71S. 1 acre, 8 perches, 15 yards. 

41 How many Irish acres, &c. in 5097 square yards ? 
Ans. 2 roods, 24 perches, 1 yard. 

. 42. How many cubic feet, &c., in 1674674 cubio 
inches ? Ans. 969 feet, 242 inches. 

43. How many yards iu 767 Flemish ells ? Ans 
07o yards, 1 quarter. 



44. How many French ells in 576 English ? Ans. 480 

.f i/\ T^J^^ J'^'-^^-' *^^" '''^^ ^^'"^^^^ of a pound 
of gold, to farthmgs ? Ans. 44856 farthin^^s 

46 The force of a man has been estimated as equal 
to what, in turning a winch, would raise 256 lb in 
?'"''?nl'/^'^ ^^' ^" ^'"^Sing a bell, 572 lb, and in row- 
mg, 608 lb, 3281 foet in a day. Uow man>^ hundreds 
quarters, &c., in the sum of all these quantities ? An^' 
16 cwt., 2 qrs., 7 lb. 

47. How many linos in the sum of 900 foet, tha 



i 



112 



nzDvvvioti. 




length of ♦ho tomple of tho sun at Balboc, 450 foot its 
breadth, 22 foot tho circuruforuiiee, and 72 feet tho 
height of many of it« columns ? ylns. 207936 
, 48. How many square toot in 7tJ0 English acres, tho 
inclosuro m which tho porcelain pagoda, at Nan-Kiiiir, 

4J. rho great boll of Moscow, now lying in a rit 
qfionnnT /"'•' supported it having boon burned, weighs 
36)000 lb. (some say much more) ; how many tons, &o , 
m this quantity ? A7i^. 160 tons, 14 cwt., 1 qr., 4 lb. 

QUESTIONS FOR THE PUPIL. 

1. What is reduction .' [1]. 
^ 2. What is the difference between reduction descend- 
ing and reduction ascending > [2] . 

3. What is tho rule for induction descending ? fal 

4. What IS tho rule for reduction ascending ? fsl 

5. How is reduction proved ? [7]. 

Qiiestiom fonmdcd on the Talk page 3, c^. 

6. How are pounds reduced to farthings, and farthincrs 
to pounds, &c. ? ^ ' t3 

7. How are tons reduced to drams, and drams tc 
tons, &c. .'' 

8. How arc. Troy pounds reduced to grains, and 
grams to Troy pounds, &c. } b j ^ 

9 How aro pounds reduced to grains (apothecaries 
weight), and grams to pounds, &c. ? 

10 How are Flemish, English, or French ells, re- 
eHs'' &o'' T ' ""'* '"''''''''' ^'^ i^lemish, English, or French 
^^11. How are yards reduced to ells, or ells to yards, 

12. How arc Irish or English miles reduced to linos, 
or lines to Irish or English miles, &c. > ' 

13. How are Irish or English square miles reduced 
to square mehes, or square inches to Irish or English 
square miles, &c. ? ^ 



n 



450 foot its 
72 feot tho 

m. 

1 acres, tho 
Nan-King, 
35000. 
[? in a j)It 
nod, weighs 
y tons, &o., 
qr., 4 lb. 



COMPOUND RULES. 



113 



a descend- 



er 



? [5]. 



S^'C. 



1 farthings 



drains tc 

lins, and 

itliccarios 

ells, rc- 
>r French 

to yards, 

to linos, 

reduced 
English 



14. Ilowaro cubic feet reduced to cubic inches or 
cu»)io inches to cubio feet, &c ^ "loncs, or 

U,,!i;-«.'':T "'•" """' ""'"™^' "• """«-■'. or naggin, .o 
lu,',»; "■"' "™ •'""'' "^""«^ '" 8"''™^ <" gal'ons to 
and';:,":: i:JX ^''^ •"■•"--) -J"-'' to pint,, 
yoa'^i &o!" ""■" ^"™ '"''"""'' '" ""^•'»- 0'- tlmU, to 
or Jw'"t:i';;;!:f:T<,.<f "■» <='™'") -^'-o'' '° tWrd., 

THE COMPOUND RULES. 

^. The Oonipound Rules, are those which relate to 
apphcate nuinbcvs of more than one denomination.' *' 
conin 1 i " ^^ '"o.u^y, wcnghts, and measures, wore 
coast, ucted according to tlie decimal sv.stcm, on v the 
T ?' ^^S^"P^« ^^^^'^'tion, &c., would be' Sir d 
to d Z \% ' ««"«i^l«'-ble advantage, and ^ a '' 
tond to snuplify mercantile transactions -If i o f • • 
things were one penny, lu pence one shillincr and To 
shil in,g. one pcnind the addition, for exan";!^ f i? 

a noun i \l \^^'- u''^- ^V'''' ^^^^^^^^ to eparate 
Id OikU r ^'" "?^ of comparison," from its parts! 

bots'follow^^^^^-"^ * " ' *^^^^^^ '' ^ P-"^)> '-Id 

1-983 
6-865 

Sum, 8-848 

The addition might be performed by tho ordinarv 
rules, and the sum read off as follows " mV . ^'^'^'"''/y 
ci^hfc sliillings, four pence, ilnd ^ 7.,^ ^^ K^'j; ^ 
even with the present arrangemoiU of mo ; wei. i 
tt^ir^ the rules alr^uly given for ad^Ji""^^' 

ttr un-/^ ' ""'f"^ "'"^^^ liave.b.en made to include 

he a d.tion subtraction, &c., of .r.nlicate .^ZZ 

can...tmg of more tlian one denomination ; sin"cc"'thQ 



l! 



¥■ 

p 



w 



114 



COMPOUND ADDITION. 



principles of both simple and coiDpound rules are pre- 
cisely the sa.no-the only thing necessary T bcnr 
carefully m n.md, being the number of any mo de! 
nomination necessary to constitute a unit o/tl next 



COMPOUND ADDITIOX. 

i^nl' nfl'^^'~^' ^i''^ ^'''7'' ^^^« ^'^^^euds so that qnantU 

vertical oIuZ ''T'T'''' "^'^^^ «^^^"'^ ^'^ ^^« --« 
wi nf ^^^•^"^n-units of pence, for instance, under 

units of pence tens of pence under tens of pence un ts 

01 shilWs under units of shillings, &c ^ ' 

U. Draw a separating line under the addends. 



aenoLi^it:^,:^^!^^^^ the same 

pence, ^., begSnin^wilh IhrLtst^^^^'""^^' ^^"^^ '^ 
ber of tl .rl? '^"\^^^;;«J c«'""m be less tlian tlie num- 
toei ot that denomination which makes one of the m vf 

^ s. d. 



52 17 33 ) 

6} } addends. 



47 
60 



5 
14 



2' \ 



IGG 17 
! and I make 3 farthings, which w^Mi ^ m..i.« r r 

-a 2 are 3, and « arc 9, aad 3 are 12"pro/i:;,'uaf tJ'S 



Ill OS are pre- 

!<'iry to bear 

any one do- 

of the next 



that qnanti- 
iu tho same 
mice, imJer 
pouco, units 

mds. 

f the same 
;«, pouco to 

n tlic nura- 
)f tho next 
ot, for each 
nomination 
one to tho 

under the 
nomination 

it in the 

47 5s. G}/1., 



ike () far- 
denoiuiiiii- 
10 jiresont. 
uarricii) 
ual to one 



COMPOUND ADDITION. Hq 

of the next denomination or thof nf c1,-it . -, 
and no pence to be sriow 1 t^r""'' ^''^ ^^"'i'^d, 
in the pence' place of the su n 1 sti r ""''z?",^ '*' ^^^P^*^^ 
and 14 are 15,^and 5 are 20 ^d 17 n i -^ i-u-^^ ''^'''^^) 
to one of tl. next ck^nomLtl or hat'of n "?.'~r^r 
earned, and 17 of the present or thnfViFn""'*'' *^ ^^ 
set down. 1 pound anf 6 l^eS^'andl are if 3' 9^ ^' 

10 pounds— equal to unit, nf Z^ j ^ . • "'"' 2 are 
1 t'u of pounds be cm ed l"^?™ ' w''" "'',''''''"■ »d 

11 and 5 L 16 tens ofTonSd^ VZ^tZT ' ""' " "" 

thoVe-aS;t;or[st r;Xt It' TT 
not so necpssflrv fn %.„+ T -i' ■•■* ^s evidently 

Of all the sunis. "^ atterwards find tne amount 

Example : 



£ 


s. 


d. 


57 


14 


21 


32 


10 


4 


19 


17 


6 


8 


14 


2 


32 


5 


9j 


47 


6 


4) 


32 


17 


2 


5(J 


3 


9 


27 


4 


2r 


52 


4 


4 


37 


8 


2 



= 151 7 11 



^ s. d. 
404 11 10. 



= 253 3 11 



a dlt .?'nfr '''^'^'''^ "'"'^ ^°^"^"' ^« "laj put down 
«rpl?-l' ""'""" ""^ '- o™- dot- "U^ing ti; S 



I 

I 






116 



COMPOUND RULES. 



& 


s. 


(/. 


67 


•14 


2 


32 


10 


4 


19 


•17 


•G 


8 


•14 


2 


32 


5 


•9 


47 


•6 


4 


32 


17 


2 


56 


•3 


•9 


27 


4 


2 


52 


4 


4 


37 


8 


2 


404 


11 


10 



2 pence and 4 are 6, and 2 are 8, and 9 are 17 pence- 
equal to 1 shilling and 5 pence ; we put down a dot and carry 
?• , ^,«:"4 2 are 7, and 4 are 11, and 9 are 20 pence-equal 
to 1 shilling and 8 pence; we put down a dot and carry 8. 
« and 2 are 10 and are 16 pence-equal to 1 shillino; and 
4 pence ; we put down a dot and carry 4. 4 and 4 are 8 and 

'T"" 1^— which, being less than 1 shilling, wo set down 
under the column of pence, to which it belongs, &c. We find 
on addmg them up, that there are three dots: Ave therefore 
carry o to the column of shillings. 3 shillings and 8 are 11, 
and 4 are If, and 4 are 19, and 3 are 22 shillings-equal to 
1 pound and 2 shilhngs; we put down a dot and carry 1. 
1 and 1< are 18, &c. ^ 

Care is necessary, lest the dots, not being distinctly marked, 
may be considered as either too few, or too many. Thia 
method, though now but little used, seems a convenient one. 

14. Or, lastly, set down the sums of the farthinfrs, 
shillmgs,_&c., under their respective columns; divide 
the tarthmgs by 4, put the quotient under the sum of tlio 
pence, and the remainder, if any, in a place set apart 
tor It m the sum— under the columu of farthings : add 
together the quotient obtained from the farthiSgs and 
the sum of the pence, and placing the amount under 
the pence, divide it by 12 ; put the quotient under the 
sum of the shilbngs, and the remainder, if any, in a 
place allotted to it in the sura— under the column of 
pence ; add the last quotient and the sum of the shil- 
lings, and putting under them their sum, divide tho 
latter by 20, set down tho quotient under the sum of 



COMPOUND ADDITION. 



117 



17 pence— 
•t and ciu-ry 
nee — equa! 
id carry 8. 
hilling and 
I are 8 and 
) set down 
. We find, 
e therefore 
d 8 are 11, 
— equal to 
id carry 1. 

ly marked, 
-ny. This 
snient one. 

farthinfi:s, 
i ; divide 
um of tlio 
set apart 
ngs; add 
lings and 
nt under 
mder the 
my, in a 
)lumn of 
the shil- 
vide tho 
sum of 



■ 



the pounds, and put tho remainder, if any, in the sum— 
under the column of shillings ; add the last quotient 
and tho sum of the pounds, and put the result under 
the pounds. Using the following example — 

£> s. d. 

47 9 21 
362 4 in 

51 16 2| 

97 4 G 
541 13 2i 
475 6 4 
6 11 11.1 

72 19 9,^ 



1G51 82 47 13 farthings. 
4 4 3 
86 50 



1055 G 2!- 



The sum of the forthingg is 13, which, divided by 4, give.i 
3 its quotient (to be put dowia under the pence), and ouh 
farthing as remainder (to be put in the sum total— under 
the farthings). 3,'/. (the quotient from the forthings) and 
47 (the smn of tlie ponoc) are 50 pence, which, being put 
down and divided by 12, gives 4 shillings (to be set down 
under the shillings), and 2 pence (to be set down in tliu 
sum total— under the ponce). 4.^. (the quotient from the 
ponce) and 82 (the sum of the shillings) are 86 shillings, 
which, being sot down and divided by 20, gives 4 pounds 
(to be set-^down inider the pounds), and G shillings (to bo 
sot down in the sum total— under tho shillings). £4 (tho 
quotient from tho shillings) and IfuU (the sum of tho 
pounds) aro 1G55 pounds (to be set down iu the sum total-- 
under the pounds). The sum of the advlendss ?s, therefore, 
found to be j£lG55 6s. 2|(/. 

15. In proving the compound rulo«(, wo can geuorally 
avail ourselves of tho methods used with the sin.i^l<? vul,^ 
[Sec. IT. 10, &c.] 



11 

m 



ssM 



(18 



COMPOUND ADDITJON. 



£ 8. d. 

70 4 6 

57 9 9 

49 10 8 

183 4 11 

£ s. d 

674 14 7 

466 17 8 

676 19 8 

627 4 2 



KXERCISES FOR THE PUPII, 

Money. 

d. £ s. d. 

7 76 14 7 
6 67 16 9 

8 76 19 10 



£ 

58 14 
69 16 
72 14 



* s. d. 
767 15 
472 14 
567 16 



6 
6 

7 
423 3 10 



£ s. 
567 14 
476 16 
647 17 

527 14 



d. 

7 

6 

6 

3 





(4) 


£ 


». d. 


84 


8. 2 


96 


4 Oi 


41 


6 






(8) 


£ 


.V. d. 


327 


8 6 


601 


2 111 


864 


6 


121 


9 84 



£ s. d. 
4567 14 6 
776 16 7 
76 17 9 
51 10 
44 5 6 



(10) 

£ s. a 

76 14 7 

667 13 6 

67 16 7 

6 4 

5 3 



2 
4 



^ s. d. 
3767 13 11 
4678 14 10 



767 12 

10 11 

8 4 



9 

6 

11 



(12) 
£ s. 



6674 17 

4767 

3466 

6r^4 

8762 



d 

16 Hi 

17 101 

2 24 
9 9 



£ s. d. £ s d 

9767 6i 6767 11 ei 

7649 11 2i 7676 16 94 

4767 16 101 5948 17 sl 

164 1 1 6786 7 6 

92 7 24 6326 8 24 



(15) 

£ s. d. 

6764 17 6| 

7457 16 5 

6743 18 04 

67 6 6k 

432 6 9 



« (16) 
£ s. d, 

634 7 114 
65 7 7 
7 12 lOi 
5678 18 8 
439 



« (17) 

* s. d. 

14 71 

677 1 

6767 2 6 

8697 14 74 

6684 0| 



(18) 

* s. d. 

5674 16 7i 

4767 17 61 

1645 19 7i 

3246 17 6 

4766 10 5| 



(19) 
£ 8. d. 

6674 1 94 

4767 11 10^ 

78 18 Hi 

19 104 

6044 4 1 



£ 
4767 14 

743 13 

7674 14 

7 13 

760 6 



(20) 
*. d. 



7i 
74 
6i 
84 
4 



5( 
6i 

34 



COMPOUND ADDITION, 



119 



(21) 
£> s. d. 
674 11 11.1 
667 14 10| 
476 4 11 
347 15 Oi 
476 18 94 



(25) 
£ s. d. 
576 4 

7 7 
732 19 04 
667 9^ 
764 2 64 



n 

6 



(22) 

£> s. d 
476 14 
576 15 

76 17 
576 11 
463 14 



7 

H 

n 

8 
94 



(26) 

* s. d. 

549 4 6i 

7 19 91 

16 64 

734 19 9i 

666 14 44 



(23) 




£ s. 


d. 


674 13 


Bk 


45 16 


74 


476 4 


61 


577 16 


04 



678 6 8| 



(27) 

£ s. d. 

876 3 

5 

66 11 11 

123 6 24 

12 





(241 




£ 


s. 


d. 


674 17 


6^ 


123 


12 


2 


667 





7* 


679 18 


91 


476 


6 


64 




(28) 


— •• 


£ 


8. 


d 


219" 





6 


32 11 


8^ 








04 


127 


8 


2 


29 


6 


6i 



(29) 



Jlvoirdupoiae Weight. 



(30) 



(31) 



"nT ^l T/-¥ ,5 ^-^-^^l- .^ cwt.qS «> 



37 
14 



2 
1 



14 44 

15 66 
11 47 



1 
3 
1 



16 
11 
16 



34 3 17 66 
37 1 16 57 
47 2 27 



3 14 

1 17 

58 2 26 



128 



12 



(33) 



(34) (35) (86) 



tT^^.t'^-iur.r^.^j.^-v-- 



66 


3 


r i» 
13 


69 


2 


17 
20 


476 
764 


3 
1 


47 


2 


17 





3 





6 


o 


81 


2 


14 


67 


1 


15 





1 



15 667 2 19 

7 4 1 20 

14 67 3 2 

18 767 1 n 



7i 
74 
6i 
34 
4 



777'- T ,■? i?.'-F%i? .T'-^^ J> -.^S 



767 
44 
567 
676 
341 



1 
1 
3 
1 
2 



16 476 "1 24i 447 

J 7 766 3 214 676 

13 767 1 16 467 

667 2 15 563 

11 973 1 12 428 



1 
1 
1 
1 




7 

6 

7i 

6 

04 



14 12 
8 4 





7 



5 








fl> 

12 
7 

15 
8 

14 



IS t,r 




130 



COMPOUND ADDITION. 



lb 
7 
6 
9 



ib 
67 
07 
66 
74 
12 



(41) 



Troy IVcight. 
(42) 



°n" *^7*- ^ff • ? °^- d^t- grs. lb 
U 6 9 6 9 - o- 

6 6 



6 



7 

8 8 





7 



7 
6 
G 





7 
4 



88 

80 





(43; 
OB. dwt. gra. 

7 9 ^8 
9 8 

8 7 6 



21 11 18 



9 
9 
8 
6 
3 



(44) 

dwt 

12 

11 

10 

6 

6 



(45) 



14 87 

11 

5 

3 44 

4 07 



oz. 


dwt. 


gfs. 


lb 


\J 


7 


12 


67 


11 


12 


3 








16 


14 


40 


12 


10 


13 


22 


8 


9 


10 


11 



(40) 

oz. dwt. grs. 

10 14 ^1 

11 :. 
9 

7 e 

18 14 




9 
8 
10 




(47) 

yds. qrs. nls. 

99 3 1 

47 1 3 

70 3 2 



Cloth Measure. 



(48) 

yds. qrs. nls. 

176 3 3 

47 2 

7 3 3 



(49) 



(50) 



y^'- *^o^- "S^' y^«- <!"• nls- 
37 3 2 2 1 



2 3 
2 



224 



6 3 2 
3 



(63) 



i,51) (52) 

UTlTTSiiifr 

54 3 673 2 3 



ts. 
*»9 
80 
98 
87 
41 



407 



1 



173 
148 

92 



1 

2 1 

3 2 



(55) 






hhds. 


gls. 


ts. 


3 


9 


89 





39 


7 


3 


40 


70 


2 


27 


44 


1 


20 


64 



TVine Measure. 

(56) 

hhds. gls. 

3 3 

3 4 

1 56 

2 7 
2 17 



(57) 

ts. hhds. gls. 

76 3 4 

67 3 44 

1 66 

5 3 4 

G02 27 



sn 



CX)MPOUND ADDITION. 



121 



(43; 




E. dwt. 


grs. 


7 9 


8 


9 8 





i 7 


6 




(46) 




. dwt. 


grs. 


> 14 


11 


11 


^• 


9 




7 


5 


18 


14 



. (50) 
as. qrs. nls. 
2 1 
6 3 2 
3 



(54) 
i. qrs. nla. 
6 1 1 
6 3 1 
110 
3 2 3 



57) 

ihds. gla. 

4 

44 

66 

4 

27 



8 
8 
1 
3 




Time. 
(58) (59) (60) 

^99 st \'- Z ff • t K'- ^f: y- ± ^-. -. 



99 859 9 
88 8 
77 120 7 



66 
67 
49 



265 115 2 42 



60 
6 





90 

76 1 

3 

1 2 



60 

67 

68 





69 127 7 
120 9 

70 121 11 
6 47 3 
8 9 11 



60 
44 
44 
41 
17 



61. What is the sum of the following :— three hun- 
dred and ninety-six pounds four shillings and two pence • 
five hundred and seventy-three pounds and four pence 
halfpenny ; twenty-two pounds and three halfpence • 
four thousand and five pounds six shillings and three 
farthings.? Ans. iE4996 IO5. S^d. 

62. A owes to B ^£567 16s. 7Jrf. ; to ^£47 I65 • 
and to D ^56 Id. How much does he owe in all ? 
Ans. iE671 12s. S^d. 

63. A man has owing to him the following sums • 

^3 10s. 7d. ■ £46 l\d. ; and ^52 14s. U. How much 
IS the entire .? Am. £102 5s. ^\d. 

64. A merchant sends ofi" the following quantities of 
hutter :— 47 cwt., 2 qrs., 7 ft, ; 38 cwt., 3 qrs., 8 lb ; 
and 16 cwt., 2 qrs., 20 lb. How much did he send off 
m all .? Ans. 103 cwt., 7lb. 

65. A merchant receives the following quantities of 
tallow, viz., 13 cwt., 1 qr., 6 ib ; 10 cwt., 3 qrs., 10 ft,: 
and 9 cwt., 1 qr., 15 ft,. How much has he received in 
all.? ^?is. 33 cwt., 2 qrs., 3 ib. 

66. A silversmith has 7 ft,, 8 oz., 16 dwt. ; 9 lb 7 
oz., 3 dwt. ; and 4 ib, 1 dwt. What quantity has he > 
Ans. 21 ib, 4 oz. ^ j 

67. A merchant sells to A 76 yards, 3 quarters, 2 
nails ; to B 90 yards, 3 quarters, 3 nails ; and to C, 190 
yards, 1 nail. How much has he sold in all .? Ans 357 
yards, 3 quarters, 2 nails. 

68. A wine merchant receives from his corj^spondent 
4 tuns, 2 hogsheads ; 5 tuns, 3 hogsheads ; and 7 tuns, 

1 hogshead. How much is the entire > Ans. 17 tuns 

2 hogsheads. ' 



II; 



122 



COMPOUND ADDITION. 



.69. A man has three farms, the first contains 120 

vf ?' on '''°^?' '^ P^'-^h^^; the second, 150 acres, 3 
roods 20 perches ; and the third, 200 acres. How much 
land does he possess in aU ? Ans. 471 acres, 1 rood, 27 
perches. ' ^.^u, ^/ 

70. A servant has had three masters ; with the first 
He lived 2 years and 9 months; with the second, 7 
years and 6 months ; and with the third, 4 years and 3 
months. What was tlie servant's age on leaving his 
last master, supposing he was 20 years old on going 
to the first, and that he went directly from one to the 
otHer .? Ans. 34 years and 8 montlis. 

nJi' Ft^ "^^°y "^^y^ ^^^"^ *^« 3rd of March to the 
23rd of June ? Ans. 112 days. 

72. Add together 7 tons, the weight which a piece 
ot far 2 inches m diameter is capable of supporting • 3 
tons, what a piece of iron one-thu-d of an inch 'in 
diameter will bear ; and 1000 Jb, which wiU bo sustained 
by a hempen rope of the same size. Ans. 10 tons, 8 
cwt., 3 quarters, 20 ib. ' 

73. Add together the following:— 2^., about the 
value of the Roman sestertius ; 7i^., that of the dena- 
rius; lid., a Greek obolus; 9d., a drachma: £3 15s 

A.d4^ 6s. 9rf., the Jewish talent. Ans. £bl\ 2s 

74. Add together 2 dwt. 16 grains, the Greek drachma: 
1 lb, 1 oz., 10 dwt., the mina ; 67 ib, 7 oz., 5 dwt., the 
talent. Ans. 68 ib, 8 oz., 17 dwt., 16 grains. 

QUESTIONS FOR THE PUPIL. 

1. "What is the difierence between the simple and 
compound rules ? [8] . 

2. Might the simple rules have been constructed so 
as to answer also for applicate numbers of difierent 
denominations.? [8]. 

3. What is the rule for compound addition ? [9]. 

4. How is compound addition proved > [16]. 

5. How are we to act when the addends are numer- 
ous ? [12, &c.] 



tlili 



COMPOUND SUBTRACTION. 



jxins 120 
acres, 3 
ow much 
rood, 27 

tlie first 
2cond, 7 
•s and 3 
ving his 
)n going 
B to tllQ 

I to the 

a piece 
'ting; 3 
inch in 
istained 
tons, 8 

»ut the 
6 dena- 
'3 15s. 
el ; und 

achma; 
7tj the 



^ 123 



ie and 

ted so 
fferent 



umer* 



COMPOUND SUPTRACTION. . 

16. I^ULE--I. Place the digits of the subtrahend 
under those of the same denomination in the minuend-^ 
tarthmgs under farthings, units of pence under units of 
pence, tens ot pence under tens of pence, &c. 

II. Draw a separating line. 



* ™- ^"^^^act eacli denomination of the subtrahend 
trom that which corresponds to it in the minuend- 
begmnmg with the lowest. 

,, ^7'J{ ^"^ denomination of the minuend is less than 
that of the subtrahend, which is to be taken from it, 
add to It one of the next higher— considered as an equi- 
valent number of the denomination to be increased : 
and, either suppose unity to be added to the next deno- 
mmation of the subtraliend, or to be subtracted from 
the next of the minuend. 

V. If there is a remainder after subtracting anr 
denommation of the subtrahend from the corro.?pond- 
lug one of the minuend, put it under the colmnn which 
produced it. 

yi. If in any denomination there is no remainder, 
put a cypher under it—unless nothing is left from any 
higher denomination. ^ 

17. Example.— Subtract £56 13s. 4^,d., from £96 75. 6|c/. 
£> s. d. 
96 7 6|, minuend. 
5613 4|, subtrahend. 

39 14 11, difference. 
We cannot take ^ from |, but— borrowing one of the 
pence, or 4 farthings we add^it to the I and then say 3 far! 
things from 5 and 2 farthings, or one halfpenny, remains • 
we set down i under the farthings. 4 pence ^om^/w« 
have borrowed one of the 6 peSce), anrone pcnny^Je 

naitpence ) 13 shillings cannot be taken from 7 but ^hor 

nC of /S^ ''T°- r ''* ^°^° 14 i^ «^e shillinl' 
place of the remainder. 6 pounds cannot be tak^n fmnf ^ 

K^y^ nave borrowed one of the 6 pounds in the minae^) 



$ 



124 



COMPOUND SUBTRACTION. 



})nt 6 from 15, and 9 remain: we put 9 under tlio units <.f 

one ol the 9), and 3 remam: we put a in tlio leas of pouud.i' 
place of the icuiamdiT. ^ 

«nnf; „T^*\^ """^^i ""*? ^^'^^ ""^"'""'^ ''f '* '^''« substantially the 
same as those already given for Siniple Subtraction [Kec. 11 
J t, &c. J It 18 evidently not so necessary to put down cvDliera 
Where there ,s nothing in a denomination of the mlluff 

suiLetrnTscc. U^Ol ''''''' '" ''' ^^"^ "^^ ^ «-'P^° 



£ «. «?. 

From 1098 12 6 
Take 434 15 8 



663 18 10 



KXERC1SK8. 



* s. d. 

7G7 14 8 
486 13 9 



£ s. d. 

70 15 6 

14 5 



£ «. d. 
47 16 7 
39 17 4 



£ .?. 

97 14 

6 15 



(> 

7 



From 



& s. 
98 14 



Take 77 15 



^ s. d. 
47 14 6 

38 19 9 



* s. d. 

97 10 6 
88 17 7 



£ s. d. 
147 14 4 
120 10 8 



(10) 

£ s. d. 
6()0 15 
477 17 7 



£ s. d. 
Prom 99 13 3 
Take 47 16 7 



« (12) 
£ s. d. 

*l^l 14 hh 
476 74 



(13) 

* *. rf. 

891 14 li 

677 15 61 



(U) 
£ v. <;. 
676 13 7^ 
467 14 92 



« (15) 
£ s. d. 
From 667 11 6^ 
Take 479 10 10^ 



(16) 

£ s. d. 

971 Ok 

7 



(17) 

£ s. d. 

437 15 

11 14 



(18) 

£ .V. d, 

478 10 

47 11 0^ 




(19) 
cwt. qrs. lb 
From 200 2 26 
Take 99 8 15 



100 



II 



Avoirdupoise Weight. 



(20) 


(21) 




cwt. qrs. lb 


cwt. qrs. 


lb 


275 2 15 


9064 2 


25 


27 2 7 


9074 


27 



(22) 

cwt. qrs. fl^ 

654 

476 3 5 






f 

ft) 

from 554 
fake 97 


COMPOUND SUBTRACTION. 

Troy Weight. 
(23) (24) 
oz. dwt. gr. It) oz. dwt. gr. 
9 19 4 946 10 
16 15 17 23 


125 

(24) 

lb oz. dwt. gr. 
917 14 9 
798 18' 17 


457 


9 2 13 







Wine Measure. 

(26) (27) (28) • (29) 

ts. hhds. gls. ts. hhds. gls. ts. hhds. gls. ts. hhds. gls. 

From 81 3 15 64 27 304 64 66 1 

Take 29 2 26 3 42 100 3 51 27 2 25 



2 52 



Time. 
(80) (31) (32) 

yrs, ds. lis. ms, yrs. da. hs. ms. yrs. ds. hs. ras 

From 767 131 6 30 476 14 14 16 567 126 14 12 

Take 4T6 110 14 14 160 16 13 17 400 15 



291 20 16 16 



33. A shopkeeper bought a piece of cloth containing 
42 yards for iS22 105., of which he sells 27 yards for 
JS15 155. ; how many yards has he left, and what have 
they cost him ^ A7is. 15 yards ; and they cost him 
£6 15s. 

34 A merchant bought 234 tons, 17 cwt., 1 quarter, 
23 lb, and sold 147 tons, 18 cwt., 2 quarters, 24 lb ; how 
much remained unsold ? Ans. 86 tons, 18 cwt., 2 qrs. 
27 lb. 

35. If from a piece of cloth containing 496 yards, 3 
quarters, and 3 nails, I cut 247 yards, 2 quarters, 2 nails, 
what is the length of the remainder } Ans. 249 yards, 
1 quarter, 1 nail. 

36. A field contains 769 acres, 3 roods, and 20 perches, 
of which 576 acres, 2 roods, 23 perches are tilled ; how 
much remains untilled ? Ans. 193 acres, 37 perches. 

37. I owed my friend a bill of £76 16s. 9i-d. out of 
which I paid £od 17s. lO^d. ; how much remained due > 
Ans. £1Q 18s. 10-^d. 



n 



■" 




126 



COMPOUND MULTli'LICATION. 



38. A norchant bought 000 salt ox liidcs, wcigliln;; 
r>01 cwt.,2 lb; of which ho sold 2r)0 hides, Vei«,'hiu3 
2.39 cwt., 3 qrs., 25 lb. How many hides had ho left, 
and what did they wci^^h ? Ans. 350 hidoH, woighin<r 
321 cwt., 5 lb. 

30. A merchant has 200 casks of butter, wcifdiing 
400 cwt., 2 qrs., 14 lb; and ships off 173 c"ask,s, 
weighing 213 cwt., 2 qrs., 27 lb. How many casks has 
ho left; and what is their weiglit .> Anx. 36 cuska, 
weighing 186 cwt., 3 qrs., 15 lb. 

40. What is the difference between 47 ]^]ngHsh miles, 
the length of the Claudia, a Roman aquoducjt, and 1000 
feet, the length of that across tiie J^ee and Vale of 
Llangollen ? Ans. 247160 feet, or 46 miles, 4280 feet. 

41. What is the difference between 980 feet, the 
width of the single arch of a wooden bridge erected at 
St Petersburg, and that over the Schuylkill, at Phila- 
delphia, 113 yarda and 1 foot in span .? Ans. 640 feet 

QUESTIONS FOR Til 12 PUPIL. 

1. What is the rule for compound subtraction ? [16]. 

2. How is compound subtraction proved ? [19]. 



COMPOUND MULTIPLICATION. 

20. Since we cannot multiply pounds, &c., by pounds, 
&c., the multiplier must, in compound multiplication, 
be an abstract number. 

21. When the multiplier doos not exceed 12 — 
Rule— I. Place the multiplier to the right hand 

side of the multiplicand, and beneath it. 

II. Put a separating line under both. 

III. Multiply each denomination of the multiplicand 
by the multiplier, beginning at the right hand side. 

IV. For every time the number required to mako 
one of the next denomination is contained in any pro- 
duct of the multiplier and a denomination of the multi- 
plicand, carry one to the next product, and set down the 
remainder (if there is any, after subtracting the number 
equivalent to what is carried) under the denomination 



i 



COMPOUND MULTIPUCATIOX. 



127 



wcigliln;* 

ho left, 
wuighing 

wciglung 
3 casks, 
asks has 
6 casks, 

>h miles, 
nd 1000 
Vale of 
280 foot, 
'oet, tho 
cctcd at 
t Phila- 
40 foot 



? [16]. 



pounds, 
lication, 



t Land 



iplicand 
e. 

3 mako 
ny pro- 
multi- 
)wn the 
nurnhfir 
liuation 






to whicjh it bolouf^s ; hut should then; bo lu) remaiudcr, 
put a cypnov in that duuouiination of the protluct. 

22. Example.— Multiply £62 17s\ lOd. by 6. 
X 5. d. 

C2 17 10, multiplicand. 
0, multiplier. 

377 7 0, product. 

Six times 10 ponce are 60 ponce ; these aro equal to '> 
phillinpfs (5 times 12 ponce) to no carried, and no pence to 
be sot down in tho product — wo therefore write a cypher in 
tho pence place of the product. 6 times 7 aro 42 shillingM, 
and the 5 to be carried are 47 Bhillings— wo put down 7 in 
the units' place of shillings, and carry 4 tons of shiUings. 
6 times 1 (ten shillings^ aro G (tons of 'lillings), and 4 (ten.s 
of shillings) to be carried, aro 10 (tons of shillings), or 5 
pounds (5 times 2 tens of shillings) to bo carried, and 
nothing, (no ten of shillings) to bo set down. 6 times 2 pounds 
are 12, and 5 to be carried are 17 pounds— or 1 (ton pounds) 
to be carried, and 7 (units of pounds) to bo set down. 
6 times 6 ('tens of pounds) are 30, and 1 to bo carried aro 
37 (tens or pounds). 

2o. The reasons of tho rule will be very easily understood 
from what we have already said [Sec. II. 41]. But since, in 
compouud multiplication, the value of tho multiplier has Jio 
cotinexion with its position in reference to tlic multiplicand, 
whore we set it down is a mere matter of convenience ; neither 
is it 80 necessary to put cyphers in the product in those deno- 
minations in which there aro no significant figures, " ^ it is in 
flimplo multiplication. 

24. Compound multiplication may be proved by re- 
ducing tho product to its lowest denomination, dividing 
by the multiplier, and then reducing the quotient 

Example.— Multiply £4 3s. Sd. by 7. 

£> s. d. Proof : 

4 3 8 29 5 8 

7 20 



29 5 8, product. 



585 
12 

7 )7028 , product reduced. 
12 )1004 
io)ou e 
quotient reduced "~4 3 8=lnultiplicand. 



123 



COMPOUND MULTIPLICATION. 



pm«'i;„'trri''hUy performed" tr"' "•,"'«■"«'«, the 
»l.ouM be equal to ,T,e ICS ' " '°™"' '"'" "'' ">'' 
ihe quantities are to bo " rednr-pfl " hoft^r.« tu^A-- , - 
gnce^the^earuer i, .ot supp'^tit' be'S :%tT"a^iJe 



1. 

2. 

3. 
4. 
6. 
6. 
7. 
8. 
9. 



s. 
9 

2 

18 
7 

13 
1 
8 

12 

17 
6 



d. 
3. 
7i 
6. 

li. 

U. 

Oi 

0. 

6. 

6. 

8. 

0. 



EXERCISES. 
£ S. d. £ 

76 14 7iX 2= 163 

97 13 6iX 3= 293 

77 10 74X 4= 310 
96 11 7iX 6=482 
77 14 6ix 6= 466 

147 13 3^X 7=1033 

428 12 7iX 8=3429 

572 16 6 X 9=6155 

,A ii^ ^'^ ^ X 10=4288 
10. 672 14 4 Xll=7399 

W ir'l l^ ^ X12=9321 u u. 

12. 7 lb at 5*. 2M #•, will cost £1 16s. Hd. 

14* ?/l'ii' ""^ W^i*^- ^' ^"1 «<^«* ^4 18,. 5J,i. 

ii'J^ra\7ii3^t.^-^hrs^^^^^^^^^ 

Rule.— Multiply successively by its factors 

Example L— Multiply £47 13s. U. by 56 
£ s. d. ^ ' 

47 13 4 

50=7x8 £ ,, ^ 

333 13 4=47 13 4x7. 
8 



T 

9 



E 



Til 
[8ec, 
tlie 1 
mult 



2669 6 8=47 13 4x7x8, or 56. 
Example 2.— Multiply 14^. 2d. by 100. 
s. d. 
14 2 

100=10x10 ,. ,;. 

£7 1 8=14 2x10. 
10 



X70 16 8=14 2x10x10, or 100. 



COMPOUND MULTIPUCATION. 

ExAMPip: 3.— Multiply ^8 2s. 4c/. by 700. 

£> s. d. 

8 2 4 

10 
— £ 



129 



81 



3 4 =8 
10 



811 13 4 =8 

7 



s. d. 
2 4x10. 

2 4x10x10, or 100. 



568113 4=8 2 4x10x10x7, or 700. 
The reason of this rule has been already given [See. II. 60]. 

26. When tlie multiplier is the sum of composite 
numbers — 

Rule.— Multiply by each, and add the results. 
ExAMPLE.-^Multiply £3 Us. M. by 430. 

£ .9. d. 

3 14 6 

10 

£ s. 



37 5 x3=lll 15 
10 



d. £ s. 
0, or 3 14 



d. 
Cx30. 



372 10 0x4=:H00 0, or 3 14 6x400. 

IGOI 15 0, or 3 14 6x430. 

r^'^^'V^nn" ?m *''^ ''""^^ ^'^ "'® «"'"° "« t^'^^t «li'e«<^y given 
[bee. II 52]. Ihe sum of the products of the multiplicand hv 
the parts of the multiplier, being equal to the product of the 
iuulti23hcand by the whole multiplier. 



EXERCISES. 



16. 
17. 

18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 

on 

27. 

28. 



£ 

3 

4 

6 

2 

3 

2 

3 







•s. 

7 

16 

14 

17 

16 

3 

4 

9 

16 



d. 

6 X 

7 X 
6iX 
6 X 



£ 
18= GO 
20= 96 
22=125 
36=103 
66=214 
64=139 
81=261 
100= 46 
X 1000=816 
100 yards at 9*. i^d. ^W, will cost £46 17 
oVa S*i||^'*^ »' ^•^*- *'f- 4K, will cost 466 13 
-4 gallons at (!.?. 8./. .W, will cost 80 
bbO yards at 13a. 4c/. 4(^', will cost 240 



X 
X 

X 
X 



s. 
15 
11 
19 
10 
8 
4 
11 
13 
13 



d. 

0. 

8. 

11. 

0. 

8. 

0. 

«.) 

". 

4. 

4 



6. 
4. 
0. 
0. 



130 



COMPOUND MULTIPLICATION. 



27 If the multiplier is not a composite nun^ci*--';-'^ 
lluLE. — Multiply successively by the factors of tho 
nearest composite, and add to or subtract from the pro- 
duct so many times the multiplicand as the assumed 
composite number is less, or greater than the giv9i-\ 
multiplier. 

Example 1 —Multiply £G2 12s. Od. by 70. 
£ s. (L 
62 12 6 
8 



76=8x9+4 



501 




9 



£ s. (I. 



4509 0=G2 12 0x8x9, or 72. 
250 10 0=02 12 0x4. 

4759 10 0=62 12 0x8x9+4, or 70. 

Example 2.— Multiply £42 3s. 4(L by 27. 
£ s. (I. 
42 3 4 
4 



27=4x7-1 



108 13 4 

£ s. 

1180 13 4=42 3 

42 3 4=42 3 



d. 

4x4x7, or 28. 

4x1. 



1138 10 0=42 3 4x4x7-1, or 27. 

The reason of the rule ia the same as that already given 
[Sec. II. 61]. ^ ^ 

EXERCISES. 

£ s. d. £ s. d. 

29. 12 2 4 X 83= 1005 13 8. 

80. 15 0ixl46= 2193 3 Oi 

31. 122 6 X102= 12469 10 0. 

82. 963 0^X999—962040 2 5i. 

2-8. When the multiplier is large, we may often con- 
reniently proceed as follows — 
Rule.— Write once, ten times, Sic, the multiplicand, 

&c., of the multiplier, add tho results. 






COMPOUND MULTIPLICATIO^f. 



131 



Example.— Multiply £47 16s. 2d. by 5783. 
5783=5 X 1000 + 7 X 100+8 x 10 + 3 x 1. 

£> •«. d. £ s. d. 

Units of the multiplicand, 47 10 2x3= 143 8 6 

10 



Tens of the multiplicand, 478 1 8x8 = 

10 



Hundreds of the multiplicand, 4780 16 8 X 7 = 

10 



3824 13 4. 



33465 16 8. 



Tliousands of the multii.Iicand, 47808 6 8x5 = 239041 13 4. 



Product of multiplicand and multiplier =27647.5 11 10. 



EXERCISES. 



33. 76 14 

84. 974 14 

85. 780 17 



d. £ .9. (I. 

4 X 92= 7057 18 8. 
2 X 76 = 74077 16 8. 
4 X 92=71889 14 8. 
7ix 122= 9013 10 3. 
7ix 162= 6865 11 lOJ. 

38. 76 gallon.^ at £0 13 4 4f , will cost £50 13 

39, 92 gallons at 14 2 4f , will cost 65 3 



36 
37 



73 17 
42 7 



4. 
4. 



40. What is the difference between the price of 743 
ounces of gold at £3 17s. lO^d. per oz. Troy, and that 
of the same weight of silver at 62d. per oz. .? Ans. 
£2701 2s. 3^d. 

41. In the time of King Jolm (money being then more 
valuable than at present) the price, per day, of a cart 
with three horses was fixed at 1^. 2d. ; what would be 
the hu-e of such a cart for 272 days ? Ans. £15 175. 4d. 

42. Veils have been made of the silk of caterpillars, 
a square yard of which would weigh about 4 grains ; 
what would be the weight of so many square yards of 
this texture as would cover a square English mile > 
Ans. 2151 tb, 1 oz., 6 dwt., IC grs., Troy. 

QUESTIONS TO BE ANSWERED 13Y THE PUPIL. 

1. Can the multiplier bo an applicato number ? [20J. 

2. What is the rule for compound multiplication 
when the multiplier does not exceed 12 ? [211. 

3. What is the rule when it exceeds ^12^ and is a 
composite number ? [25] . 



m 



i!li 



132 



COMPOUND DIVISION. 



4. When it is the feum of composite numbers ? [261 
6 When It exceeds 12, and not a composite number ? 

6. How is compound multiplication proved ? [24]. 



COMPOUND DIVISION. 

29. Compound Division enables us, if we divide an 
apphcate number mto any number of equal parts, to 
asceitam what each of them will be; or to find out 
anrth'^r^"^ ^'"'''^ """^ applicate number is contained in 

If the divisor be an applicate, the quotient will bo an 
abstract number— fur the quotient, when multiplied by 
the divisor, must give the dividend [Sec. II 7yl • but 

ronV^^^ri"!!' ?>'^^''' ''^"'"^* ^« multiplied together 
L20J. If the divisor be abstract, the quotient will be 
applicate— for, multiplied by the quotient, it must give 
*he dividend-an applicate number. Therefore, either 
tlivisor or quotient must be abstract. 

ceeTl^^^"^ *^'^ ^^^'^'^'' ^^ ''^^*''^«^' ^^^ <ioes not ex- 

RuLE— I. Set down the dividend, divisor, and sepa- 
rating Ime— as directed in simple division [Sec. II. 72]. 

II. Divide the divisor, successively, into all the deno- 
minations of the dividend, beginning with the highest 

111 Put the number expressing how often thl divisor 
s contamed in each denomination of the dividend under 
that denomination— and in the quotient 

tinnTf tL* r •^''''',°' ''^'^^J contained in a denomina- 
tion of the dividend, multiply that denomination by tho 
number which expresses how many of the next lower 
denommation is contained in one of its units, and Tdd 
the product to that next, lower in the dividend. 

wn V * ..A^^^^l ^""^^ succeeding remainder in the same 
way, and add the product to the next lower denomi- 
nation in the dividend. umomi 

VI. If any thing is left aftor thn nnoti-nt ^"o-- -^i- 
lowest denomination of the dividend is obtained; pi^t'iJ 



COMPOUND DIVISION. 



13: 



in 



I 



down, with the divisor under it, and a separating lino 
between : — or omit it, and if it is not less than half 
the divisor, add unity to tho lowest denomination of tlie 
quotient. 

'61. Example 1.— Divide X72 6s. did. by 5. 

£> s. d. ' 
5)72 6 



14 9 



4i 



5 will go into 7 (tens of pounds) once (ten times), ami 
leave 2 tens. 5 will go into 22 (units of pounds) 4 tiraos. and 
leave two pounds or 405. 405. and (Ss. are 4Gs., into which 5 
will go times, and leave one shilling, or lid. 12t/. and \)d. 
are 21tZ., into which 5 will go 4 times, and leave Ir/., or 4 
farthings. 4 farthings and 2 farthings are 6 farthings, into 
which 5 will go once, and leave 1 farthing— still to be divided ; 
this would give \, or the fifth part of a farthing as quotient, 
which, being less than half the divisor, may be neglected. 

A knowledge of fractions will hereafter enable us to 
understand better the nature of these remainders. 

Example 2.— Divide £52 4s. l^d. by 7. 

& s. d. 



7)52 4 



n 



7 9 2 
One shilling or \2d. are left after dividing the shillings, 
which, with the Id. already in the dividend, make 18(7. 7 
goes into 13 once, and leaves 6rf., or 24 farthings, which, 
with f , make 27 farthings. 7 goes into 27 3 times and G 
over ; but as G is more than the half of 7, it may be consi- 
dered, with but little inaccuracy, as 7— which will add one 
farthing to the quotient, making it 4 farthings, or one to 
be added to the pence. 

32. This rule, and the reasons of it, are substantially the 
same as those already given [Sec. II. 72 and 77]. The remain- 
der, after dividing the farthings, may, from its insignificance, 
be neglected, if it is not greater than half the divisor. If it is 
greater, it is evidently more accurate to consider it as giving 
one farthing to the quotient, than 0, and therefore it is proper 
to add a farthing to the quotient. If it is exactly half tho 
divisor, we may consider it as equal either to the divisor, or 0. 

33. Compound division may be proved by multipli- 
cation — since the product of the quotient and divisor, 
plus the remainder, ought to bo equal to the dividend 
[Sec. TI. 79]. 

g2 



134 



COMPOUND DIVISION. 
EXERCrsKS. 



1. 

o 



90 



7 



7(5 14 
47 17 



10 
11 



3. 
4. 
6. 
6. 
7. 

8. 97 14 

9. 147 14 
167 16 
176 14 



96 19 
77 10 
32 12 
44 16 



d. £ 

6-1. 2=48 
7-j- y=25 
0-^ 4=U 
4-:- 5=19 
0=12 
7=4 



7-i- 
o 



ft. d. 

3 9. 

11 6i. 

19 U. 

7 10.^. 

19 5i. 

13 '3. 



7 -f. 8= 5 12 l" 
17 1|. 
16 5i 



3-^ 9=10 
0-4-10=14 
7-Ml=14 
6-j-12=14 



6 
14 



11) 
6i 



The above quotients are true to the nearest fa. dnl^. 

number!?''' '^' ^^"^^''^ "^'^^^^ ^^' ^"^ ^^ ^ compos-it. 

Rule.— Divide successively by the factors. 

ExAMPLt.-Divide £12 175. U. by 36 
3)12 17 9 

This rule will be understood from Sec. II 97. 



12. 



EXERCISES 

£> i. d. £ ^. 

J* II 6-^ 24= 1 



d. 

81 



13. 676 13 34- 36=16 4] 

14. 447 12 2-5. 48= 9 6 6 

16. 647 12 4-^ 60= 9 i? 7 

17. 740 13 4-5- 49=15 2 sj. 

Jte n^W- ^^"^ '^'"''^ 12, and ia no. a oom- 
h„fi!i!''^7^™'"""^, ""y *''<> ""ttod of Ions division • 

set down tlie inultiplic-a, fco obt-inM "' «"'aond, 
tient as directed in long'dH.ision 'tSo II.'S]! '^'"" 



■"^ 



iposite 



lom- 



lon ; 
tiers 
eno- 
iiext 
[)nd, 
j[uo- 



t;OMP0U.\D DIVISION. 135 

ExAMPLK.— Divide £87 IG.v. 4d. hy G2. 

£> s. (I. £, ,<. f/. 
62)87 10 4 (1 8 4. 
G2 

25 

20 multiplier, 

shillings 5 16 (=25x204-10) 
490 

"20 
12 niultiplior. 

pence 244(=20xl2-f4) 
186 

~58 
4 multiplier 

farthings 232 (=58x4) 

180 

"40 

C2 goes into £87 once (that is, if gives £1 in the nuoti.nir), 
ard leaves £25. £25 are equal to 500.s\ (25x20j, Avhich, 
with 10.S. in the dividend, make SlCs. 02 goes into 51fis'. 8 
times (that is, it gives 85. in tlie quotient), and leaves 2(»i-., 
ov 240^2. (20x12) as remainder. 02 goes into 240, &:c. 

Were avo to put I in the quotient, the remainder would ba 
40, which is more than half the divisor; we consider tlm 
quotioiit, therefore, as 4 farthings, that is, we add one penn^ 
to (3) the pence supposed to be already in the quotient. 
£1 8*'. Ad. is nearer to the true quotieut than £1 8s. 34Vi.[32]. 

This is the same in principle as tlie rnle given above [30]— 
buv since the numbers are large, it is more convenient actually 
to set down the suras of the dilVorent denominations of the divi- 
dend and the preceding remainders (reduced), the products oJ 
the divisor and quotients, and the numbers bv which we multi- 
ply for the necessary reductions : this preveuta the memory 
from being too much burdened [Sec. II. 93]. 

36. When the divisor and dividend are both applieate 
numbers of one and the same denomination and no 
reduction is required — 

FtULE. — rroeeud as already dii'ected fScc. II 70, 
72, or SU]. ' 



136 



COMPOUND DIVISION. - 



Example.— Divido £45 by £5 

£5)45 
Tliat is £5 is the ninth part of £45 

nominatiou is found in cither, oi' both^ '"' ^'• 

ItULE.--Ileduco both divisor and dividend M fl.n i 
est denomination contained in eithor m / .u^ ^^"^ 
ceed with the division ^^^' ^^^ *^^^^ P^^ 

ExAMPLE.--Divide £37 5.. 9ld. by 3.. Ghl 
'' i'. £ s. d. ' ' 



3 
12 

"42 
4 



170 farthings. 




5 9J 



M 



170)35797(211 
340 



179 
170 

"~97 



Theroforo 3.. Qid. i, (ho 
211th part of je37 5y.9,V/. 
*J7 not being less than the half of 170 ^91 ^^^ -i . 
OS equal to the divisor, and therefore addSf J fT Tr''^?'' '* 
as the last quotient. ^"^^eioie add 1 to the obtained 



18. 

19. 

20. 

21. 

22. 

2-3. 

24. 

25. 

26. 

27. 

28 

29 

SO 



£ s. 
176 12 
134 17 



1736 


14 


73 


16 


147 


14 


157 


IQ 


68 


15 


62 


10 



8764 4 

4728 1;} 

8204 

5236 2 



EXEHCISES. 

d. 

2 -^ 

8 ~ 
7 -^ 

7 -f- 
s -^ 
7 -1- 
2 -i. 
6i^ 
Oi~ 
2 -f- 

H-i- 
7|-^ 

2 —9842: 



• 191: 
183: 

443= 

271: 

973= 
487= 
751 = 
419= 
408= 
317= 

261:= 

875= 



£ s. 

■ 18 



d. 

6. 

14 9. 
10 13 104 
" 6 
3 
6 
1 
2 



5i. 
5|. 




= 
= 
= 

: 

:18 14 

:14 18 

31 10 114. 

■■ 5 19 81 
9 4]. 



111. 

Gh. 
41 



COMPOUND DIVISION. 



137 



licato, but 
a one de- 

) the loT? 
then pn. 



31 . A cubic foot of distilled water weighs 1000 ounces 
what will be the weight of one cubic inch ? Ans 
253'lS2i) grains, nearly. 

32. How many Sabbath days' journeys (each 1155 
yards) in the Jewish days' journey, which was equal to 
ii3 miles and 2 furlongs English > Ans. 50-66, &c. 

33. How many pounds of butter at ll^d. per lb 
<vould purchase" a cow, the price of which is £14 15s. ? 
Am. 301-2766. 



QUESTIONS FOR THE PUPIL. 

1. What is the use of compound division .? [29]. 

2. What kind is the quotient when the divisor is an 
a])stract, and what kind is it when the divisor is au 
applicate number ? [29] . 

3. What are the rules when the divisor is abstract, 
and docs not exceed 12 ? [30] ; 

4. When it exceeds 12, and is composite ? [34J ; 

5. When it exceeds 12, and is not composite ? [35] ; 

6. And when the divisor is au applicate number ? [36 
and 37]-. 



'■ i^ lh( 



isidor it 
)btainoJ 



13S 



Sii:oTioN jy. 

FRACTIONS. 

?n<l one TZZ'^CtZlf^'f.''''" ^I""' P"'«. 
is called a /„„«« ""''° P«'« »'•'> token, we have what 

re,t';4:r;';:;'Lt3e-St'%'''!,.''--- "- been 

la a fraction of 5— its siv(l, 1 . !i •'^ '' f^""- H- ««!) 
into six e,,„al part f w n ?tn ' *""" ""'Z boingdivid i 
wo shall sJ-o prLen iy) »' un?^^™?,^ *'""» i "' (»» 

4 Srchtgo^fcl^ZV"^^^^^^^^^^ » fr- 

or, while the Jeuom „'uoV tell the ^' ''"■mimtor ,■ 
kinci of parts into which the nn;,- *"'"'"««*« or 
divided, the numerator "m„!t/« J, '"^f"""^ '» ""o 
number of them which TtTn t!"' "l f*'"""'^ 'l-^ 
sevenths) means thottL ! ,/''"M (read three- 

" 'tree- V"dr'ttXm''ed ^^r''^'" ""^ "-' 

'Trei;::?;t:'/» ^~th?f::r:.-» -^ 

of the fraot! nX~h™ .'• '^^ ?'°'" ""^ ™'" 
JiviJc the numeSor bv thT"J'™' "^'"^'^ *'"=" we 
value ; and the Kreater .l,r^^ •T"]"'"''''' '' "« ■''^al 
quotient. On the^eomrart .h7*",'* '¥ ''■"■§«"■ «'o 
♦or the less the frae on^:,-' T''^ ""^ denomina- 

'1- smaller the quTt „t7seo jf 7^^^^^^^ "^"-^ 
greater than 4— whieh ;« i? i T®^ =— hence 4 is 

nun.r'atrr^ ?dXls^r' V'''^^-«-' "' »' 

^l-«-'i-neha,^^d.1l^X;r^lSn"-S 

• -^ xro ean 

no 



FRACTIONS. 



139 



to increaso or <liinini«U both the (Ji\ iJciiJ and divisor — 
which does not afFoct the (diotieut. 

5. The following will rc^. resent unity, seven-sevenths, 
and live-sevenths. 



I 



I U"^^y- I Lii 



?i 



The very faint lines indicate wliat ^ wants to make 
it equal to unity, and idmlical with ^. In the diagrams 
which are to follow, we shall, in this manner, generally 
subjoin the difference between the fraction and unity. 

The teacher should impress on tlie mind of the pupil 
that he might have chosen any other unity to exemplify 
the nature of a fraction. 

6. The following will show that ^ may be considered 
as either the -f of 1, or the | of 5, both — though not 
identical — being perfectly equal. 

\ of 5 units. 



? of 1 unit. 



Unity, 



DIDIl 





















m 








1 


\fi\f 









































o\ 



Tn the one case we may suppose that the five parts 
belong to but one unit ; in the other, that each of the 
five belongs to differeii ' units of the same kind. 
^ Lastly, ^ may be considered as the | of one unit fivo 
times ua large as the former ; thus- 
I of 1 unit. 




I of 5 units. 



equal to 



■I .. 
.1 . -. 



0\ 



"•* 



HO 



FnACTIONS. 



at leapt, of the quotiont^nUo'Llntotr '"'''''•' ''^''' 

not. uknHcalZk''Z , j^lf C.S'T*"V!"' ^' 
lowmff will exomnlifv +].! • "'^^"^" h iho foi- 





thus J i3 evidently eqS to 'r"*' *f f »''■">»"*»■■ ; 
noticed when we trite'd of divisfo^ Lril.Tl/'"^"^ 

tbo>;e4eteabt";7t\"n;tL"° '"''™ ".^Z^'' 



KKACTION«. - I4j 

11. A cnmpnwnd fraction siipposos ono fraotioa (o 
refer to another ; thus J of J— roproscntiid iilso hy J X ^i 
(throe-fourths multiplied by four-ninths), means not 
tho four-ninths of unity, but the four-nintlia of tho 
throo-fourths of unity :— that is, unity boia,^' divided into 
fjur parts, tlirco of thcso arc to bo divided into nino 
parts, and then four of those nino arc to bo taken ; thua 

I ? 




o 



TIT 



I r 



12. A complex fraction has a fraction, or a mixed 



number in its numerator, denominator, or both : thus t, 

4 
which means that wo are to take the fourth part, not 
of unity, but of the ^ of unity. Tiiis will bo eicm- 
pufied by — 




8 ?- U U 







- 








3 

r 


1 








f<:cj^ 




J 


1 




^ 











u 

7' s' 
1 -4 

5 



— , are complex fractions, and will be better 



54- 



understood when we treat of the division of fractions 

13. Fractions are also distinguished by the nature of 
thou' donommators. When the denominator is uniiv, 
foUowed by one or more cyphers, it is a decimal frac- 
iion—imL^, /it, f Ao , &c. ; all other fractions are vilo-ar 
^thus,i,|,73_, &c. 

Arithmetical processes may often be performed with 
fractions, without adnalhj dividing the numerators by 
the denominators. Since a fraction, like an integer 
may be increased or diminished, it is capable of uJli* 
tion, subtraction, &c. 



142 



FRACTIONS 



niinaticm!' '''^'^' """ "'^'°"' ^° '' ^''''^^^'" '^" ^^^ ^^"«- 

uni^v"" !r.^T' "^^ ^ co^'^idercd as a fraction if wc make 
unity it^ denominator .-—thus f may be taieu for 6 • 

if 21° '""^ ^7" ™, '."J^S"!- any denominator we please 
If we previously mnltiply it by that denominator ; 

tims, 5=1 , or f , or f , &c., for 22=5X5_5_5 . 
, 30 5X6 5 ' ^ 1X5-1-5, 

and -— = =-=5 &« 

6 IXQ I ^ 

EXERCISES. 

^i S^'^''''^ '^ *° ^ ^'■''''*'''°' ^^""'"S ^ ^' denominator 
tor^" I'f."' o^^ *° ^ ^'^'*^'"' ^^"^^"S ^^ ^« denomina- 

^',i^^'' l^^'- 1?=V. I 5. 42= V/. I 6. 71 = «F« 
lo. lo reduce fractions to lower terms 
Before the addition, &c., of fractions, it will be often 

Trirplpe- ^^ *'^^^ *^^"^^ ^^ ^^^<^^ - P-^^^- 

■n, 40 5 40 40-i-8 <! 

Example.— =j^=-. For ~— 3^^Xr_^ 

72 9 '^^'72-72-^8-9• 
We have already seen that we do not alter the quotient- 
which 18 the real value of the fraction [4]-if we XSv ot 
divide the numerator and denominatoh)y the Tame number 
he^atr:' ^''" ''''' ^''- "• ''"' ^^^^ ^^ usefull^rem , : 



Reduce the 

7 -574 387 

10. MJ% 
12 ?§CFr' 

In the answers to 
in future, generally 
mlnations. 



EXERCISES. 

following to their lowest terms. 



13. 
14. 
15. 
16. 
17. 
18. 



Fa 3 1 • 

4 3 4 

Fll 3"* 

60 5 

12 S"- 

.98 7 

1 1 2 — y 



19, 
20, 
21. 
22. 
23. 
24. 



100400 1004 

7^00 2' 

5UO0 1120 

firnj — t:?ii' 

4 2 5__ s's" ' 

1^^, — I?T' 
30 8 



4 12. 



-"BTr="-2n-3 

5I2_2 5 

B J i — SoT 



questions given aa exercises, we shall, 
reduce fractiona in fhoJi. In^nof d Z_ 



FRACTION*. 



143 



ny deno- 

we make 
I for 5; 

e please 

5 



nmator 
lomina- 



8514 
»4 ' 



16 often 
ossible, 

ommon 



tient— 
tiply or 
umber, 
•emein- 



— 1004 

-3ffoa- 

1 20 

?T1' 



h' 



shall, 
^ — 



28. £-'=ir).v. 

29. £-j^-=5.s-. 

30. £^|„=1./. 



1«.'|^ find the value of a fiactiou in terms of a 
lower a6iaomination — 

RuLR.^;^rteduce the numerator by the rule already 
given [Sec. III. 3], and place the denominator under it. 

ExAMPLK.— What is the value, in shilUngs, of J of a pound ^ 
£3 reduced to 8hillings=60.s. ; therefore £'i reduced to shil- 
liiigs=';j^s. 

TJie reason of the rule is the same as that already given 
[Sec. III. 4]. The | of a pound becomes 20 times as much if 
the " unit of comparison" is changed from a pound to a shilling. 

We may, if we please, obtain the value of the result- 
ing fraction by actually performing the division [91 • 
thus \°s.~los. :— hence £^ = l5s. * 

EXERCISK.S, 

25. £U=Us. 6d. 
2G. £1^=1 7s. 4d. 
27. ^ll-^lOs. 

17. To express one quantity as the fraction of an- 
other — 

KuLE. — Ecduce both quantities to tho lowest deno- 
mination contained in either — if they are not already 
of the same denomination ; and then put that which is 
to be the fraction of the other as numerator, and tho 
remaining quantity as denominator. 

Example.— What fraction of a pound is 2W. ? £1=9G0 
earthings, and 2,|./.=9 farthings; therefore ^fr^ is the re- 
quired fraction, that is, 2^(1. =£y^^. 

om '^•^Tt^ ^*' "^^""^ lluLE.— One pound, for example, contains 
JfM fartlnngs, therefore one farthing is £-.' (the OGOth part 
of a pound), and 9 times this, or 21, is £9 X y|o=ffT!r- 

EXERCISES. 

31. What fraction of a pound is 14?. 6d. ? Ans. ^a 

32. What fraction of £100 is 17^. 4d. ? Ans. -ia°." 

33. What fraction of i^lOO is i:32 10^. ? Jm-.*i|! 

34. What fraction of 9 yards, 2 quarters is 7 yar*ds, 
3 quarters ? Ans. .ai. *^ ' 

35. What part of an Irish is an li^nglish mile > Ans n 

36. What fraction of 6s. iid. is 2s. Id. ? A,is. X ' 
ov. What part of a pound avoirdupoise is a pound 

Troy? Am. -m. ^ 



u^ 



«.' 




VULGAR FRACTIONS 
QUESTIONS. 



1. What is a fraction ? [1] 

.. 7/ >Vhj may the numerator and denomhiatnr he. «,„i 

o. What IS an improper fraction ? [71 
b. What IS a mixed number ? [8] 

numbejTrg]"" ""^^P^' ^'■^'"^"^ '"^uccd to a maed 

denomination? [14]. ^ *^ "^ fraction of any 

^^12. How is a fraction reduced to a lower term? 

a ^Jt2^^^^ ^-^- ^-^^ - terms of 
of ItotfeT' tl7T ''^'''' '"' ^"'"'^^^ ^« the fraction 



VULGAR FRACTfOiNS. 
ADDITION. 

18. If the fractions to ho Tddo/I i,n,,^ 
denominator— ^^ ^^'^^° ^ common 

Example. — a 4. g — i t 

Reason of THF- Tfrrrw T^ -I, 



if 



VULGAR FRACTIONS. 



145 



tbeir nature. 






Unity. 






1. 3-K+^=v==i5. 

3. |.'4-'"4.iL=;!o_o4 

O. f ^-U'-*_l_lo 4 2 03 



EXEnciSES. 

I t), l^J_ll_l_l«_33_0 

10. -lT4.'t!-f.iii_.5_^_.oin 

12. ^\-f a 4.11^3^^1 ri? 

19. If tlie fractions to bo added Iiave not a'c'ornni..n 
a|«ator, and all the denonnnato.s are p iL^rS 

Example.— What is the sum of 5-f 34.4 ? 
|+^+ ^=.2X4X7 3x3x_7 4x3xi ^6 63 48 m 
H • ;^t^'^X^X^^7x3x4=84+84+84=8T 

factors^ (tlio ' v^Mdenomnnt T"' "''/"'' ^ ''^"«« ^'^« ^=""« 
the same product '^'"""^^'^^^^^ors) mu«t uecemrilj produce 

confmof trmrn'aoT we^'w""''' ^^"^ "'^ ^^^^--'^ *« a 
rator and denomi air of e^^ "Ct/^^'^'P"''^ ."'« ""'"«' 
[4] does not alter the fraction "^ Tf- '^'"^ ""•"^'''•' ^^'^<^^ 
common denominator; for 7f L „ ],/%r''T'''-^: ^^ ^""^ <* 
out so doin,, w. cannot pil ^1^ tloililaJTri^-^n^it 
of them as tlie denominator of tlieir sum;-thus ^^±^+^ 

So ^uSffi^r t jiii^ns^r .:? r ^ -^^'^ ' ^^^ 

and sevenths. Avhich avo Lo! 1, J'"''P."'^ ^''^"» "»« <onr(l,3 



N-^+i 



are less t) 



be correct — 



tl.an thirds; noiihor wuukt 
«ince it would suppose nJi ,<y tU«m to be 



f 



146 



! 1 Tilt , 



lis 



VULGAR FRACTIONS. 



equivalent number of others wl.icl L"e ,„,„ll,r 1 ' '° "° 

H+?=2_X4 3X3__8 9 17 
_ 3 4 8X4'^4X3~12+12"=J2 

These fractions, before and after thev r..r.fMvn .. 
denommator, will be represented as follo'y^s :- '"""'^ 




8 








1 


(' 



equal to 






■ w iai n an^JL *— 



as w: ilr diss ihd/^i^^^f °' ^'^^ r? j-^ - --'« 

than twelfths, we eouk not C« f ''!,"*'^ ^"'^'" I^'"'^'^ l'"-^'««- 
exactl,oquivalenl"res;!ectitely:iot?AS^r '' '''"' 



12- 3+f+f 



EXERCISES. 



20 



=5 9 139 



22. If the 

denomiuutor, 

to each other- 

Procoed as 

Rule. — Fi 

denominators 




fractions to be added have not a common 
and all the deuommators are not primo 

directed by the hist rule ; or— 
"d the least common multiple of all the 
[Soc.II.l07,&c.|,thiswilllieth.cnnnnnn 

muifciply tno numerator of each fraction 



8n_i 5 



i coniiiiou 



VULGAR FRACTIONS. ^4-* 

into the quotient obtained on dividing the common mul- 

tiple by Its donominator-this will give the new nume- 

vators ; then add the numerators as already directed [18]. 

ExAiMPLE. — Add X -I- A -I- 3_ OQQ • .1 , 

■^^^ 3 2 1-4^5 -f-^ 2- Z6ii IS the least common 

multiple of 82, 48, and 72 ; therefore --i.l.i_288-i-82x5 
.281-f:48X4 288^72X8 45 24 ' if 8^"""^^^"" 

ii88 + 288 =288+2T8+288=2T8- 

multiple of the denomiLts^K^Tnt ^6^si'JT'''' 

__ 5><288 ^ ^^""^ 

instance)- g^q-^. For we obtain the same quotient, whether 

diminish the numher^f^^^^^^^^^^ 

tionsTe oS.lr h^'''''^'"' ""? ""'^ P""^° *'' «^«I^ other the frac 

jS>?e ff ^e a:::^:z:::^i^j^t^]^^-:^ 

denommators, the common denominator irthe present n 
stance, had wo proceeded according to tho L, ru^e ?9], wi 
would have found 1 , j. , j. __ 17280 18432 4608 
4032£ . ^^^ 40320 ^^^^"^2-110592+110592+110692=* 
110692 ■ " 110592 ^^ evidently a fraction containing larger 

terms than --. 

288" 



25. ^_1_SI4.^14 3 O'JS 

2b. 44.^_Lj__t;7 9." 

27 ?TfTf >2— -r2.- 



EXERCISES. 



^4-5 i 4_.:jii.3__2-4 7 



34. ;v?|+|=fCl^ 

35. |flflL45i?'' 

37. |+HJ!f!fe:Z:?;T^7 



»2T0"' 



28. l+l-Li^^dllZ^Vl^- 

29. iJiJ niiST^^- 

30. |I|t^=,i?Z{y 

tioni^'" ''^'''" ' ""^^'^ ^^"^^^^ to"iu" improper frac- 
ha^n.''';il^!?"^%*^° '^^'^'''^ ^'^'^ i«i« a fraction, 

te.^:dn;tri^L.^,.r ^-^-^ p- 

^^Ex^PLK.-What fraction is equal to4f ? 4^ = 4_f.5=, 



Mi 



II '. 



148 



VfLUAU FRACTIONS. 



-o. Reason- of thk Rirr p _ \Va i „ i •. 
integer may bo expre;sc. us a ^r u Hon f '"'''-^ '''^ '^'^' ^ 
nator >ve pksase :-the reduction nf„ •'"'T'''" ''"^ ^^'"^""- 
fore, is really the addition of Z.r '"'''''' ""mhor, tliere- 
a coiiiiuou deuomiaator. ^^''^"tions, previously reduced to 

EXERCISES. 

44. 99A.=:iono 

45. 12''.i=>i^ • 

40. i4Lo/: ■ 



47. 40 -'==3 73 

48. 13 =1 o; 

49. 27||=iv. 



38. 16'=''3^ 

39. 18|=i|o. 

40. 79.=«|3. 

41. 47 ='|3. 

42. 741=017. 

43. 95^=^f6. 

26. To add mixed numbers— 

KuLE.--Add together the fractional parts- th^n if 
«ho sura IS an imnroner fr-^ot;..^ T P. ^^' ^'len, if 

■="»"■« 1-What is the sum of 41 + 18 " 



7 I 5 12 14 

«T^8 — 8 =J-i 



sum 



isl 






eighths-that ; , one o^t a;Hed''^n;: .**; T '"''' ^'^ ^ 
and 18 are 19, aid 4 are 23 ' "" "'*^ ^"'^"- -^ 

KsAMPLE 2.— Add 12f and 29i.l. 
|4-|-|==4l=l.i 7 12^=12.3 5 



3 

sum 42'-I 
y 



T„ .1 • . . sum 4z.'ri 

rW an,r 22f t'XTS l^fj H'''""""*? ""^ -Edition 
aenominatol^ ""* ''^"'"""^ l^to to ,v oomiuon 

beS-is"ptSir,?er™'il"-*|"'» ^' "'-0 '-™- 
but, in the first exa™t.efi,ril?''''? f ""P'" "M'ion 
Won is equal to „S oTthl „" . ST'-"*^' ''.'' O"" <>™"mi„a. 
.1. 8], J« Of one dealttr„'^*!S '^ TeTf ttt"." ''^° 

50. «+3|=8S. '"^"'-■'"='- 

61. 8f4.4.2|l==lltRi 
52. l65'H-fOG^L^"' 



53. 10+11^.^22^ 



54. 



II MS 






56. 4r+3§j:H6 =n |»- 



58. 92.4.+37Ji+7t-ifi7J5, 

59. 17aA+8!3tl|T=27^||J. 



VULGAR FRACTIONS. 



149 



th.it an 

dcnomi- 

'', tliere- 

(iuced to 



if 



set 



QUESTIONS. 

1. Wliat is tho rule for adding fractions which have 
a common denominator? f"18]. 

2. liow are fractious brought to a common denomi- 
nator ? [19 and 22]. 

1 ^' S** ^^ *^^ ^"^^ ^^^' addition when the fractions 
have different denominators, all prime to each other ? 

|_lc)J. 

4. What is tho rule when tho denominators are not 
the same, but are not all prime to each other ? [22]. 

5. How is a mixed number reduced to an improper 
fraction ? [24] . ^ 

6. How are mixed numbers added ? [26]. 

SUBTRACTION. 

28. To subtract fractions, when thcv have a common 
ionommator — 

^ .Rule.— Subtract the numerator of the subtrahend 
ifrom that of the minuend, and place the common deno>. 
iftimator under the difference. 

Example.— Subtract -J from J. 

7_4_7--4_3 
9 9~ 9 ""9' 

20 IlEAsoN OF THE lluLE.— If We takc 4 individuals of anv 
Kind, Irom t of the same kind, three of them will remain In 
the example, we take 4 (ninths) from 7 (ninths), and 3 are loft— 
wluch must be nintlis, since the pvoces.s of subtraction cannot 
have changed their nature. The fullowin- will exempUfy the 
Bubtractiou of fractions :— a ^ ni>uiy me 









^^ 



Unity. 



7 
IT 



w 


r 


73 


4 


iiii 
















i 





a> 

I 



H 



Uil'i' 



I 



II i'lr 



Ihif 



160 



VULGAR FRACnONS. 



1. 

2. 
3. 
4. 
5. 



EXERCISEa. 



— ^,=t.. 




c. 

7. 
8. 
9. 
10, 



1* « . 5 



U-XAMPLE.—Subtract ^ from |. 






ft! T> " " -^2 i"5=^tf. 

OX. JtRASOIV OF THP T?tttw tx • 

g;^ven f20] for reducing fractbns !« !''"''''' *" *'^'*^* ^^^^^'^^^ 
previous!; to adding them ^'^ "^ '^^'^'^o" deaominator'. 



11. f- 

14. Il-i^ll^s 

'* 13 — Iffy. 



EXERCISES. 



_S 7 

_S — 39 

"8' 



I 



15. 

16. 
17. 
18. 



11^^13X_ 769 

11-4 HZ's'^^** 
4 8 8i — -o-g. 

75H_320_2;! 



m^fd n'umberf'' '"^^^ "^°^^^-' ^^ frrctions from 

iVitfrsXL^^^^^^^^^^^ :.---" denominator- 

W from that of the ifnd 1'.'' ^f.^ ^' *^^ ««^*^- 
ence with the commS^ Zn • /'* ^^^^ *^« ^^^ff^r- 

subtract the integTpart ofT k'. T^'.' ^'^ *^^" 
integral part of the minueL subtrahend from the 

that of'Ih^:/£t nd, Clt r r '' ^- *^- 
mon denominator to ts nu2r./ ^ .'^'"^ *^*^^ «^°^- 
mtegralpartofthemi^uendT;^^^^^^^ '"' '^^^''-^^ *^' 
Example 1.~-4| from 9|. 

9f minuend. 
4| subtrahend. 

^ /»• vxi. y. ^*^ difference. 

from &'^S,4!'«''"'' -d 2 eighth. (=.; ,,„^, , 



.VULGAR FRACTIONS. 



151 



not a coin- 

inator [19 
rule. 



at already 
uominator. 



7_e9 



2" 



ns from 

inator — 
subtra- 
e differ- 
!; : then 
•om the 

ss than 
com- 
ise th'i 



ka. 4 



ExAMPLK 2.— Subtract 12.^ from 18]. 
18j minuend. 
12^ subtrahend. 

5^ difference. 

3 fourths cannot be taken from 1 fmirth ; but (borrowing 
one from the next denomination, considering it as 4 fourths, 
and adding it to the 1 fourth) 3 fourths from 5 fourths and 
2 fourths (== j) remain. 12 from 17, and 5 remain. 

If the minuend is an integer, it may be considered as 
a mixed number, and brought under the rule. 

Example 3.— Subtract 3f from 17. 

•,JJ i!?^^y ^® supposed equal to 17^; therefore 17-34=3 
17^-3^. But, by the rule, 17^-3J=16|-3f =13^. 

83. Reason of the Rule.— The principle of this rule is 
the same as that already given for simple subtraction [Seo 
11. ly] :— but m example 3, for iustance, five of one denomina- 
tion make one of the next, while in simple subtraction ten of 
one, make 07ie of the next denomination. 

34. If the fractional parts have not a common deno- 
minator — 

Rule. — Bring them to a common denominator, and 
then proceed as du-ected in the last rule. 

Example 1.— Subtract 42| from 56^. 
56|==56y*^, minuend. 

subtrahend. 



42 > = 



-42X 



I'lyV' dilTerence. 



85. Reason of the Rule.— We are to subtract the dif- 
lerent denommations of the subtrahend from those which cor- 
respond m the minuend [See. II. 19]-but we cannot subtract 
iractions unless they have a common denominator [30]. 



EXERCISES. 



19. 
.20. 
21. 
22. 
23. 
24 






15|-7|=7 
12f— 12 
8411 _lt_; 



941 



14iif— |f=1473-. 

24. 82iH-7iif=74e. 

25. 762-72/^^3 Jf."^ 



26. 67|-34X=32^1. 

27. 971-32J|=64-TC 

28. 60|-4ll(=19i! 

1001— 9|=9ni' 

60— A=59,« '■ 



29. 
30. 
31. 
32. 



12|-l01= 






t 



u-^ 



i'l y 



I6d 



TULQAR FRACnONS. 



.hi 




I" '^ 
|i 



QUESTIONS. <^ 

9 W»,o* • ^u ^^'""^on (ienouiinator ? [281 



MULTIPMCATION. 
tholonLy^""'^ * '■'•''°"»'' l-y » ''hole number; or 

ExAMPLE.—Multiply f by 5. 

37. ReASOIV of the TfTTn? T 1 . 

we are to add the multS^ca^ r«-" T?^^oi*^ ^^^ ""'"ber. 
as are indicated by the mu t?pSer?^but ILf?/' V^^^^y times 
a common denominator we must add ftn ""^^ factions luaving 
put the common denominatorTn'dfk^o'^rX^^^^^ -^ 

whTchcoSSe'tL^Scin'^o'^^^^^ "P^^^'«"'«f *hV integer 
multiplier-their s"zeSTunoZZ"''n '^P^?f "^ ^^ *h« 
be the same thing to incrlR7fh!?l^ • l^ ^^^^^ evidently 
Without altering %l,eirS'^!^S,lT ', \" ^^"'^l «^t««t 
dividing the denominator bvTh?l-'^°"^'^ ^^ ^^^^t^d by 
AX5 = -|. This will become Jll. ^'''° -i^^^^^P"^^ 5 thus 
the fractions resulting W Uf C£^t^o^otfr3^-^^^^^^^ 
common denominator-for ?? /=lX6\ ^^^ 4 •__ 4 ^T 
will then be found equal ^^ ^ ^^ '^' '''' ^ (""15X5) 

denornl7or'Srnu£ber'o?f-*'P"^^ ^^ not contained in the 
the method given in th?rf/;!- """' expressed by an integer 
, The rule wiirevUntly anSv""?/'.^'- T"^ ap^plicable ^''' 
Pl>ed by a fraction-s"fce^£ Ir^^ '""^T^ '' *« ^e multi- 
whatever order the factorrart'tn? rr«L'1"'^^i^ '^''^'^ i» 



VULGAR FRACTIONS. 



153 



38. Tho integral quantity wlilch is to form oiio of 
the factors may consist of more than ono deuumiuatiou 
ExAMi'LE.— What is tho f of £5 2s. \)d. i 

£ 8. d. k 5. d. £ s. d. 



5 2 Ovj ^5 2 9x2 _3 g 



0. 



1. fX2=l|. 

2. 5x8=6^. 

3. f,Xl2=:10J. 

4. Jxl2=91. 

5. VVx30=14. 



11. i2x«G=34. 

12. i«x20=l9. 

13. 22x|=4f. 

14. AXI7=U 

15. l43xH6i|. 



EXERCIBKS. 

6. 27x1=12. 

7. Axl8=3«. 

8. 1|X8=71. 

9. 21xiJ=9. 
10. 15x1=3. 

16. Plow much is -^^ of 26 acres 2 roods .? Ans 
20 acres 3 roods, 

17. How much is \^ of 24 hours 30 minutes } Ans 
7 hours. 

18. How much is /jVa of 19 cwt., 3 qrs., 7 & .? ul7w 
7 cwt.,3qrs.,2 1b. 

19. How much is if of dC29 } Am. £\y ==£Q 195 

39. To multiply one fraction by another — 
Rule. — Multiply the numerators together, and under 
their product place the product of the denominators. 
Example.— Multiply | by |. 

4 5^4x5 20 
9 6 9x6'^54- 
40. Reason of the Rule.— If, in the example <,nv..n, wo 
were to multiply f by 5, the product (^^O would be tinea 
too great— since it was by the siMh part of 5 (^), wo should 
have multiplied. -But the produ.;t will become wliat it ought 
to be (that IS, G times smaller), if we multiply its denominator 
by b, and thus cause the size of the parts to become 6 times less. 
_ We have already illustrated this subject when explaiu- 
mg the nature of a compound fraction [11]. 



20. 
21. 
22. 
23. 



■LvS 3 5 



l^Xt!" 



^ ^ l^A. — -'8' 
I v^ .1 



x^= 



EXERCISES. 

XV4 4 8 



48 1 



24. 1^X4! 
25.JXfXfV=^5 



ffS- 



27 



3 14 vl?7. 



3^3 
si) ?.4 



28. 
29. 
30. 
31. 



32. How much is the %■ of 3. > y} ,,? i 



X V 1 1 

fjXA=fV 

I 2 -^ 8 — 1 6* 



-fins. -i. 



33. H ^w much is the f of f ? A //y 



t ; ' 



■f^"' 



IM 



Vl/LOAR FRACTIONa. 



of a fraotinn W "' "• . ^e8»"«8, the multiphcat oa 

that of division ; and the number said in hi . u-v^i 
must bo made loss than boforr ^' multiplied 

mkfd n^u"^^^ ' *'^^^'^"' ^^ ^ ^'^ ---^er by a 

raf/tn'T;?^*^"'^ '"''','^ °""^^«^« to improper fractions 
r24J, and then proceed according to the last rule 
Example 1.— Multiply J by 4|. 

"*!==*? J therefore ^ x 44=A y * > ~ 1 2 3 
Example 2.— Multiply 5J bv 63 *'^" 

52=V. and Gf=3^- tterefore 52x6f=:V x^^=^^^ 

„. EXERCISES. 

37, Ax8Iy» v.. '■=;,.. £' Pixl3|x6|= SoTiJ. 

^ 44. m. is t.o p.oa„et of e:'^1tt77> 
^45.^ What i. tho product of f of f , and | of 3f » 

8 4' # 



r 



f 



. ULOAR FRACnONS. 



105 



44. If wo perceive the numerator of one fniction to 
bo tho same as the denominator of the other, wo may, 
to perform the multiplication, omit the number which 
m commou. Thus f X5 = f. 

«.;'!" 51* '* ?f, *""'® f 'livJding belh tl.e numerator ami fbM.o- 
mmator of the product by the saiuo nuiubuv— uud theicfore 
does not alter its value; since lut^itioia 



o^y= 



5 



-ti~i}' 



4.). Somctnnes, before performinju; the mnltiplic.-.tion, 
we can reduce the numerator of one fnicti.>n nnd I ho 
denominator of another to lower terms, by divi.Iin ' 
both by the Ranio number :— -thus, to multiply i by •» ^ 

Dividing both r; and 4, by 4, we get in their placos, 

d and 1 ; and the fractions then are A and J which 

multiplied together, become -^-X 4 = 7'-. " ' 

tnJni-^/" *^'^'''''P« as dividing the nuincrutor and denomina- 
tor of the product by the sanio number ; for 

8^7-bx7-j-4-2x'7 V=2^7/ ^H' 

QUESTIONS. 

1. How isafrnction multiplied by a whole number 
or the contrary ? [36] . 

2. Is it necessary that the intccror which constitnfos 
one of the factors should consist of a single denomina- 
tion.? [38]. ° 

3. What is the rule for multiplying one fraction by 
another ? [39] . "^ 

4. Explain how it is that the product of two proper 
tractions IS less than either .? [41]. 

5. What is the rule for multiplying a fraction or 
a mixed number by a mixed number ? [42]. 

6. How may fractions sometimes be reduced, before 
they are multiplied ? [44 and 45] . 



166 



VULGAR FR ACTION'S. 



WVlSIOxY. 

tl.o whole 3^ and tui;';"'","' "'" «■''*""" h 
inerator. ' I^"' ""= Product under it, „u. 

Example. — ?--i-4 ^ ^ 

47. ReASOJV of THF TJirrr. Ti t ., 

for instance, is to make ifSr '''"^''^^^ '"^ quantity hv 8 
^t is evi.leut that if, wwl L i lo .?'^''"'^' *''"'^" ''^'^'"ro- J3 t 
same, we make theiTlhA If^'' \^'^ '"'"'^'''' ^'^ *''« P'lrts the 
Jteelf 3 times Jes.s- !Urct to muU! f ''..^^ T^^^^^ theSvact a 
is to divide the fraction bv tL i-,, ^'"^ * '^ denominator by 3 
A similar cifect wT k^ f """; ^^umber. '-^^ ^' 

^iule we leave the .^. of^Sp^t" ;^ ^ "^^^f!. --|"-| if. 
--^.. 3 times less; thus '--^iJ^/J "^ "'^'^^ ^^-- 
numerator is not nh„ '^ ", '^ ~>' "^'"^ ^"'^° *^'« 

of -^ comi.l,i lZfoT[l'2] " '^'^ ''^P'""'«i ">o nature 



1. II_i_9 4 

2. |4^^"7 



4- ^-9=^. 



ESERCISK3. 



«• H-8=,, 



12. -^,-M4_/''^^ 



7. A->14= • . 

wlieu we multiply rdtido?s'° ^'^ ? "^^^^or, th.-,,f, 
nator-bj the sa no r mm or w ?""^*^^'^l«^ ^"d denonu-' 

«nce we then a tlTZl r ""^ ^^^'^■^" ^^^^ ^^^1"«- 
dcoroase it. ' '" '"'"" *^^^> ^^I^^^^J increase and 

4f'- To divide a fraction hy a fraction- 
llur,E.— Invert the dlvf^ri /"''^'^^^n— 

'^o^ed), and then placed ^i/tL7^r ^' '^ ^^ ^^■■ 
multiplied. ^ " ^"^ fractious were to bo 



VULGAR FRACTIONS. 



157 



Example. — Divide | by f . 

5_^3_5 4_5x4_20 

7*4 7^3 7x3~2r 
Reason of the Rule. —If, for instance, in the 
just given, we divide | by 3 (the numerator of the 
we use a quantity 4 times too great, since it is not I 
the fourth part of 3 (|) we are to divide, and the 
(£y) is 4 times too small. — It is, however, made what 
to be, if we multiply its numerator by 4 — when it 
1^, which was the result obtained by the rule. 

50. Tho division of one fraction by another may bo 
illustrated as follows — 



example 
divisor), 
y 3, but 
quotient 
it ought 
becomes 






5 


. 


3 

'.A 








«n 


•a 


* 




ns 


- 


— 



The quotient of f-r-f must be some quantity, whic-b, 
taken three-fourth times (that is, multiplied by -^-), will 
be equal to f of unity. For since the quotient multiplied 
by the divisor ought to be equal to the dividend [Sec , 
II. 79] , -f is f of the quotient. Hence, if we divide tho 
five-sevenths of unity into three equal parts, each of 
these will be owe-fourth of the quotient — that is, precisely 
what the dividend wants to make it four-fourths of the 
quotient, or the quotient itself. 

51. When we divide one proper fraction by another, 
the quotient is greater than the dividend. Nevertheless 
such division is a species of subtraction. For the quo- 
tient expresses how often the divisor can be taken from 
the dividend; but were the fraction to be divided by 
unity, the dividend itself would express how often the 
divisor could be taken from it ; when, therefore, tho 
divisor is less than unity, the number of times it can bo 
taken from tho dividend must be expressed by a quantity 
greaUr than the dividend [Sec. II. 78] . Besides, divid- 
ing one fraction by another supposes tlie multiplication 
of the dividend by one number and the division of it by 
another — hut when the multiplication is by .a ffrea';er 



4 



m 



'hfl 



fW 




•4 2 



158 



VULOAU FRACTIONS. 



must beicrea™!, " ''""""'^ ""^ '» •"> -iviJod 



13. ^-^3=I^! 

14. 4-^2=1 

15. 14-1=13. 



EXEnClHKS. 



?7- l"^H'- I i»--fi-^=i^' 
18. j#-i=ii. 21. ?j.jrr 



62. To divide a whole number by a fVaotion' 

minator of the product '" ''' '''''''''''''' '^^^ ^eno- 

Example.— Divide 5 by ^ 

5^3^5x7^a5 

' 7 o "-" "5" • 

This rule is a consequence of Hm i..of . ^ 
t-cr may be considered a" "fr- t\ V •'' '^"'-^ ^'"^^^ """i" 
iDinator [14]; hence sl" J'^n^'x-- "'^ """''^^ ^'"' '^'""- 
b^'<S; dJn^SS U!'' '^''^'^ --'^-' «'-11 consist of 

Example. -Divide 17.s'. ShJ. }>y 3 



22. 3-4-1=6?. 

23. ll^/*.=,i.'u 

24. 42-;;j,=gG4. 



I-\')SRCISES. 

25. 5-i-i|=,')i 



31. Dmde £7 IGs. 2d. by a yi., o,. , / ^ , 7 

32. Divide ^8 13.. 4d by r J ' f n J"^^'^ 
3.Prv^e^5 0..1..by^r j^;; ^^^ ^ 

o^:S. 10 divide a mixed nmnbnr h,r o T.ri i , 

or a fraction— '^•^ ^ ^'^^^^^ number 

and 49]. ■'' ^''''^'' '' ^^^"^^^ directed [40 

Example 1.— Divide 9f by 3. 

9?-^3=9-f-3-f■;^ -^3=34-1=3' 
Example 2.— Divide 14^3- by 7 ^' ''' 

'.4''='*^'' therolore lM--^2=',F-M = '^X^='?-« 



'■■.■^n:'^:-'^0^m^f ''■:■''' 



.5 

.'J 



VriLrjAR FU ACTIONS. 



159 



54 llEx^oN OF THK RuLE,~lji the first cxnniplc we have 
divided each part of the dividend by the divisor auu added 
the vesults-which [Sec. II. 77] h the same as dividing the 
^•liole dividend by the divisor. 

In the second example we have put the mixed number into 
a more convenient form, without altering its value 



EXERCrSES. 



34. 8|^17=f|. 

35. 51|-^.3=lK\ 




'151 



rs- 






.1^ fl450 
■''Tf. 



39 433S_:_41 , 

40.- iifi^MSffir 

41, 18-8X^^5— 19*773 

42 loVJ^irliim, 

43. 18±-Ail=ii«7 •'"^ 



'■v^' 



55. To divide an integer by a mixed number 

Rule.— Reduce the mixed number to an improper 

fraction [24] ; and then proceed as already directed 

1 52]. 

Example. — Divide 8 by 4§. 

41=%^ therefore 8-f.4f=8-^ 2^3=8 x./j^Uf. 
Reason OF THE RuLE.-It is evident that the" improper 
fraction which is equal to the divisor, is contained in the divi- 
ilend the same number of times as the divisor itself. 



' 1 






44. 
45. 

4S. 
49. 



46 14-^l|=7^-V. 

47 21^ll3*-=ii-i 



. 3 
5^- 



n^d. 



EXERCISES, 

5-f.3l=lf, 

16-i- 1112—113 3 

Divide £7 16s. Id. by 3i. Ans. £2 6s. 
Divide £3 3s. 3d. by 4i, Ans. Us. O^d. 

56. To divide a fraction, or a mixed number, by a 
mixed number — "^ 

ron'''''''75'''^''''' "''''',^ numbers to improper fractions 
L-J4J ; and then proceed as already directed [49]. 

Example 1.— Divide | by 5 J. 

6l=.f, therefore | ^5^=^^5/X,3><^^^,^^ 
Example 2.— Divide 8j9j- by 7|. 
8A=fJ-, and 7|=V, therefore 8X-^7#r^"_..y-r.7 w 

47 Reason of the Rule. — We (n.<i in tim loof •, i \ , 

ehango tho K,ixed nvmbcr, into *o h e" fm J ' cot S'j 
dmcled-without, however, altering their value °'"'"'"'"'"'' 



o 11 

fl 56 2 



'1 

i 



^ij 



160 



VCLGAR FKACT/ONS. 






50. ~'--^.TJ. — •lii 

51. ^14-41=11* 

52. J<.^3A=Vi3 

53. U^jlll^¥^^- 

54. MJ-sCfT 

2 • ^3 ^32. 



EXERCISES. 



55. 82 rV— 26 /'-=<= '?«■■■ 
50 ioM\i-^*'',ti<'>^^- 

57. ^^:^8^=::i^3»^^^"'"--- 
59. 2|J.34+i^e-..h, 



58 When the divisor, dividend' o/hnfJ. " 
poand, or complex fraetimis- ^ ^'^^'' ^'"'^ "^""- 

Wrsr^'p^Xm^'ft^ -del,, to sin,.e 

which are cLCuTL^d tt 7^ ^« ^^'"'^^^ 

are complex ; tLntr'ocTe'd ^^l^^S^Z^^:^) 

^Example l._Divide 4 of | by f . '" ' ^"''^ 

f of «=3o {-39j^ ^j^^^^^^^^^ fX«lo=-^n_no^._,,, 



Example 2.— Divide ~ bv •' 
i=4% M, therefore |-g=^^.=^x|=^, 



o 



rn ^ o EXERCISES. 

01. 4fi-l— S^VJ* «;n43 



62. 



'8~ IT * 



21 

2 •> 



63. H^sy_7_n7 
97 • 3'^i5— im- 



o4. . — 1- S.~r-on 

UO. — _i;.2'v « OfTt 

19 '" 

<i 

60. ~_i.2v5 Q231 

7 

QUESTIONS. 

I. How is a fraction dived bv an integer ? r4fil 

o. J^ixplam how it ocputq f^.;* *i """.• L'*-*J. 

fractionsis sometimes greZthal t fT' ?i '^" 
4. How ;.! -, »i,,i 8'"'"" man the divider,-. J fsi i 

[62]. ' " "'""^ """'''«'• divided by a finetlW? 

ggj' number, by a mi^d number ? [55 and 

to.I- r'l;tVot oSpTe;t^^^^^^^^^^^ [fir. " 



.*^Jr:.. 



.'■^ -„ :t^ .= 



VULGAR FKACTIONS. 



16J 



MISCELLANEOUS EXERCISES IN VULGAR FRACTIONS. 



acres, 3 roods ?* 



Ans. 



1. How much is ^ of 1S6 
20 acres, 3 roods. 

2. How much is ^ of 15 hours, 45 minutes ? 
7 hours. 

3. How much ir, //f% of 19 cwt., 3 qrs., 7 Bb .? ^7w. 
7 cwt., 3 '^rs., 2 tb. 

4. How much is ^V/^ of £100 ? Ans. £3Q 95. 

5. If one fanii contains 20 acres, 3 roods, and 
another 26 acres, 2 roods, what fraction of the former 
is the latter > Ans. -^^\. 

6. "What is the simplest form of a fraction express- 
ing the comparative magnitude of two vessels — the one 
containing 4 tuns, 3 hhds., and the other 5 tuns, 2 
lihds. ? Ans ' " 



JlS. 

o n 



1. What is the sum of | of a pound, and f of 



a 



shilling ? Ans. 13^. lOfri. 

8. What is the sum of |-5. and ^ul. ? 

9. What is the sum of dCi-, a/ 
36'. If-irZ. 

10. Suppose I have -^ of a ship, and that I buy y\ 



Ans, 7j\d. 
and y'jf/. } Ans 



more ; what is my entire share ? 



Ans. J-i. 



11. A boy divided his marbles in the following manner : 
he gave to A ^ of thorn, to li j\, to C i, and to D i, 
keeping the rest to himself; how much did he give 
away, <and how much did ho keep ? Am. He gave away 
tVo of them, and kept j\?^. 

12. What is the sum of | of a yard, j of a foot, an(J 
4 of an imh .' Aiis. 7 inches. 

■'3. Wha. is the difference between the | of a pound 
and o}d. ? Ans. lis. 6^d. 

14. If an acre of potatoes yield about 82 barrels of 
20 stone each, and an acre of wheat 4 quarters of 460 
lb — but the wheat gives three times as much nourish- 
ment as the potatoes ; what will express the subsistence 
given by each, in terms of the other ? Av:<. The pota- 
toes will give 41 } times as much as tlie wliL^a, :, and tho 
wheat the ,£j\ part of wliat is given by the poiu-toes. 

15. In Fahrenheit's thoniiDnioter "there arc 'SO de- 
grees between tho boiUng and freezing points , in that 






163 



iniClMAL rUACTIOiVS. 



of Rcaumav only 80 ; what fraction of a do.rcc in tho 
lat or oppresses a cjegrcc of the K,r,ner r X? ' " *^'' 

isak.'nt ia -^T^' ^'^ 'f '''''' ^" t^^« United" Kingdom 
w about 34 inches jn aepth durino- tlio voir in fl,o t ,• 

of hi boii /!'" """"":>"'''. "■' 22,480 ; what fnlctiou 
1. «i^ "* ^ '"■S"""! ''^l"™^"^ tl'^'-t of Chimbora.o ? 

jjoi, w^i r » r zr To=x™rri 

fraction of the latter ? ^w^. f.i. "^ '-^prc.ssca as a 



DECIMAL FlliVCTlONS. 
59. A deun.:il fraction, as ah-eady rcmarlied Tl-^l 

Sinoo the division of\Su™l;Cof\t ci:!! tc'l™; 
by .ts d,3n„™.„to,-fro,„ the ve..y natu,ror ,!^ ^ S 
infnf fl J""'^ Pra-formod by moving the do.nmil 

60 It is as inaccurate to confound a decimal fraction 
with the corresponding decimal, as to confoZd a 4w 
fraction with its quotient.— For if 75 i< th. fi , 

from eit,-:^"""™* °' ^ "'• "' ^^- "'l Ciually'distior; 

mnl 'f. ^."<"^™»I « cliangcd into its correspoudi,,.. deci- 
ml fraction by p„tting unity witl. as manv cypfcs 4 

f»"^ .';'''"" 'r^' "■'' '!=«'"■■>■: point. Thus -Oe^fi^ 

I O 1) ) '^^ 



,•5 (1 4 r, 

I J 



DECIMAL FnACTI0N3. 



163 



gi'Go in tlio 



I Kingdoiu 
the plains ; 
mt fractioa 

Iiigh, and 
at fractiou 
limborazo ? 

fissure or 
bet as tlie 
lountairis ; 
sscd as a 



:ed [13], 
lilt hand. 

fra<.ition. 
I fraction 

nutation 

deoiuial 
ho equi- 

facility, 
ty by a 
nt three 

fraction 
a vulgar 

q.wtknt 
; so also 

distinct 

ng deci- 
)hers as 
nator — • 
5646= 



^2. Decimal fractions follow esactif UiC same rules 
as vulgar fractions.— It is, however,* generally nioro 
convenient to obtain their quotients [oG]^ and then per- 
form on them the required processes of addition, &c., 
by the methods already described [Sec. II. 11, &c.] 

63. To reduce a vulgar fraction to a decimal, or to a 
dediiial fr actio, i — 

IluLE. — Divide the numerator by the denominator- 
tins will give the required decimal ; the latter may be 
changed into its corresponding decimal fraction— as 
already iokicribed [61]. 

ExAMPLK 1.— Reduce I to a decimal fraction. 

4)3 

Example 2.— What decimal of a pound is lid. * 

7^/.= [17] £i^- but £5Vo=^C-0032, &c. 
This rule requires no explanation. 



EXERCISES. 



1 7 _8 7_5 

3. ^V— 36. 

4 i' — jK'L 

^' 4 100' 



5. •! 



625. 

6. -^^=-973&o. 

7. J=-5. 



9. -j«,/.,=-90476, &c. 

10. |=.8. 

11. /^=5625. 



13. Ileducc \2s. Gd. to the decimal of a pound. Ans 
625. 

14. Reduce Ids. to tlie decimal of a pound. Ans. -75 

15. lleduce 3 quarters, 2 nails, to the decimal of a 
yard. Alls. '875. 

16. Ileduce 3 cwt., 1 qr., 7 lbs, to the decimal of a 
ton. Ans. -165625 

64. To reduce a decimal to a lower denomination . 

IluLE. — Ileducc it by the rule already given [Sec. 
III. 3] for the reduction of integers. 

I^lsAMPj.K 1.— Exjjress £-6237 in terms of a shilling 

•6237 

20 



. li 



ft 



'in 

11 



Answer, i2'4740 shillings=.CG237 



B! 'if 



HI-. 



164 



DECIMALS. 



Example 2.-Rcducc i;.9734 to shilling. &c 
•9734 
20 

19-4G8() 8hIllings=X-9734. 

5-G160 ponce=-4C85. 
4 



2-4040 favthinffs=-G10^Z 
Answer, X-9734=las. 5 J,/ 

ro^S-oI'^^^^^^^^^^^^ a, wc. given 

of » shilling by rrroJuccs rt ■^'"'''l,''^ "8,'l'° <1««"""1 
J'Wmy. MultiDlvinff t^ryj!, ? '^.t"™ ""'' "'" ■)«"»"'>' of a 

EXERCISEa 

23. WJuat is tlie value of £-80875 ? yl,,,. 17,. 4^,1 



-4. What IS the value of £-d375 ? 



20. I ow much is -875 of a yard. ^ ^M,s^L^^^^^^^^ 

27. AVhat is the value of £-05 ? Ans. Is 
^^f^- How much is -9375 of a cwt. ? Ans. 3 qrs., 

29. What is the value of £-95 } An^. \Qs. 

30. How much is -95 of an oz. Troy ? Ans 19 dwt 

31. How much is -875 of a gallon > Ans 7 nints 

28'%'':^.r''' ''-'''' ^''^ ''^y^ ^1-// hours, 

;J^' !^!'^ f^"owing will bo found useful, and—befnr, 
n nnatcly connected with the doctrine of SctionT^ 
may be advantageously introduced here : '^''^^^^"^-- 
io imd at once what decimal of a pound is enn.V. 

Whor;l"""'^" «f ^^^^. ponce, i?r '^""" 
\V on there is an even number of shillings- ' 

a pound" "''^'' '^''''' '' ^' ^'''^^ ''' ^^^y t^^^tbs of 



DECIMALS. 



165 



ivcre given 

i it to shil- 
l»o decimal 
cinial of a 
I'educcs it 



175. 41(1 

■ , 2 nails. 
• 3 cwt., 



3 qrs., 



19 dwt. 
pints, 
hours, 

ins. 1 5 

— being 
tions — • 

c-qniva- 



?tlis of 



E.VAMrLE. — lC)S.=£-8. 

Every two shillings areciiual to one -tenth of a pound; there* 
fore 8 times 2s. are equal to 8 teuths. 

67. Wbon tlio nunil)er of shillings is odd — 

lIuLK. — Considor half tho next lower oven number, 
as so many tenths of a pound, and with these set down 
5 hundredths. 

Example. — 15.s'.=£-75. 

For, 15.? — 14S.+1.S. ; but by the last rule 14s.=cC-7 ; and 
Binco 2s.=l tenth— or, as ia evident, 10 hundredths of a 
pound — l.s.=5 humlredths. 

68. When there are pence and farthings— 

K,uLE. — If, when reduced to fjirthings, they exceed 
24, add 1 to the number, and put the sum in tho second 
and third decimal places. After taking 25 from the 
number of farthings, divide the remainder by 3, and put 
the nearest quantity to the true quotient, in the fourth 
dechnal place. 

If, when reduced to farthings, thoy are less than 25, 
set down the number in the third, or in the second and 
third decimal places ; and put what is nearest to one- 
third of them in the fourth. 

Example 1. — What decimal of a pound is equal to 8,J(L T 

8J=35 farthings. Since 35 contains 25, wo add one tc ■ 
the number of farthings, which makes it 30— we put 36 in 
the second and third decimal places. The number nearest 
to the third of 10 (35-25 farthings) is 3— we pvit 3 in the 
fourth decimal place. Therefore, 82=£03G3. 

Example 2. — What decimal of a pound is equal to 1:^(?. 1 

1^=7 farthings ; and the nearest number to the thir*! of 
t is 2. Therefore l|tZ.=£0072. 

Example 3. — What decimal of a pound is equal to 51(1. "? 

5](/.=21 farthings; and the third of 21 is 7. Therefore 
- 3](i.=i:0217. 

69 Rkason of the RuL,E.--We consider 10 farthings as 
the one hundredth, and one farthing as the one thousandth of 
a pound — because a pound consists of nearly one thousand 
farthings. This, however, in 1000 farthings (takeu as so 
many thousaudtlis of a pound) leads to a mistake of about 40 — 
Binco ,fil=(not 1000, but) 1000—40 farthings. Hence, to a 
tliousaud favihings (considcveil as thousandtliB o^ a pound), 



ii 



/ /THs: .\i 



* '4 




1C6 



11 at' 



CIRCULiTlJtG UE^.jiAl.,. 



correction "liouhhtill bo I,, , M ,";;•■ "• " "'''"'" 

or .«.«w one, i„ t,,o/„°!;?raii;f„,'fr;s;'' ''* 

KXERCISEfl 

18 7 i ft. ==:£ -0822. 

J9. ^-'7 5*. I0rf.=je27-2916 



tljo •ne 
tiiat, ua 

must be 
nunibor, 

that the 
number, 



20. 4,,.3|,/.=£.7i55 

^-i. £42 ll5.6R=i;.42677 



^-, in an/dtilrora'X^i:::!^"- "^ *"'»o-. ponce, 

tJonsMer the digit in ttc™?! ?,,'«" '',""'' °™- 
6, If It 18 not loss thau 5^ r,„„i /f''''' ^n'>'™o'ing 
;«.unito of farthings ; and s, b ta 1 "" •, f' '" ""^ "'"-^ 
>f It exceeds 25. *"'" "'"'^ f™ni the result 

ExAMrLi:.-je-6874=I3s 9rf 

4r^^^t^::^^^:^i^^ -«- widths 

tiie remainder (reduced fn f i, ^^"n^/redtJis and adding 
sandths, we have^T thousandtt r^^^^^^l^- *^ *he tS 
exceed 25, we subtract Svtfe i™ ^vhich-since they 
of farthings. ^-6874 theJefoS i T' ?^/^ *^« ""™be? 
tarthmgs-or 136-. 9d. '"^^®^"^®' i« ^qual to 13^. and 3G 

"- -^^ ^^"owa fro. t.e I.t three^being the reverse of 



|;:|i 



CmzVLATim DECIMALS 

''w»yi;ra"f 'c "ottro?^ ?^*'?'' [See. H. 72], 
number by a„otlor?-ll ST' ".''° '""''^ ™« 
'ta..Iy recur, 0^0^,^ a'^^-^E' T *'^'"'; ™"- 



CIRCIJI,ATI.V(; BE(,...IAL8. 



Ul 



docimal is i)roclue( il.— Tho decimal ia euiJ to bo termi- 
nal, ,'.' there is an exact i|uouciit — or oue which loaves 
no romaiiider. 

72. An iutormiiiato dcciiiKil, in which only a ingle 
figure is repeated, is callc ' a rppettnd; if two or more 
^ligits constantly rccnr, thoy tonn a. periodiai/ '--' \, 
Tiins ;77, &c., is a repcfnnd ; hut -59759/, ..o. i.s a 
periodical.^ Vov the siko ot brevity, tlie repeated digit, 
or period is set down but once, and may he marked as 
follows, -5' ( = '555, &c.) or -M'jri' (-- 193493493, &.c.) 

Tlio ordinary method of marking i\n: period is sonie- 
wliat different — what is liore given, howcv<T, seems 
preferable, and can scarcely be mistaken, even by those 
in tlio habit of ^ -^ing the other. 

When the d imal contains only an vnjivile. M-t — 
that is, only tlu; repeated digit, or pei-iod — it is u pure 
repctcnd, or a p?/. rc_ periodical. ' But when there is hoth 
a finite and an infinite part, it i.-s a mixed repetend or 
viixe.d circulate. Thus 

j*'^ (=V)oo, &c.) is a pure rcpotcnd. 
'578' (=-57iS88, &c.) is a niixod ropotond. 
'397' (= o97397;">97, ka.) is a pure circulate. 
8G5^G427r(='8G5G427164271G4271,&c)is a mixed circulate 

73. The number of digits in a period must always ba 
less than the divisor. For, different digits in the perioci 
suppose different remainders during the division ; but 
the number of remaind;MS can never exceed — nor even 
be equal to the divisor. Thus, let the latter be seven : the 
only remainders possible are 1, 2, 3, 4, 5, and 6 ; any 
other than one of these would contain the divisor at 
least once — which would indicate [Sec. 11. 71] that the 
quotient figure is not sufficiently large. 

74. It is sometimes useful to change a decimal into 
its equivalent vulgar fraction — as, for'instancc, when in 
adding, &c., those which circuhite, we desire to obtain 
an exact result. For this purpose — 

IluLE — T. If the decimal is a pure rcpclevd, put the 
repeated digit for numerator, and 9 for dcnominato.-. 

II. If it is a 2>ii''& puriodimly put the period for 
numerator, and so many nines as there are di<;its in the 
period, tor denominator. 




H 




IMAGE EVALUATION 
TEST TARGET (MT-3) 





i: 






■V'J- 

•ip M^ 




^ A' 



W- 



W^r 



/ 







^ 



1.0 



1^ 

1^ 


m 


\m 






142 


1^ 

136 


22 


\L. 


liiH 





I.I 



1^ RRffil 



1.25 



12.0 



1.8 



14. ill 1.6 



V] 



<^ 



% 



/a 



/a 











w 






om 



9 



V// 




Phntnoranhir 

Sciences 
Corporation 



23 WEST MAIN STREET 

WEBSTER, N.Y. 14580 

(716) 872-4503 





V 



iV 



^ 




^^ 





\ 



\ 








A 






6^ 



>^ 



m 



'' ' - ,""■ > »'* 



S"y ■ . 



„"-;S,' >. iC'"" 



IG8 



CIRCULATING DECIMALS. 



^^ Example l.-What vulgar fraction is equivalent to -2' 1 
Example 2. — What 



*7S54'? Am. 



7S54 
VW99- 



vulgar 



fraction is equivalent to 

we multiply two equal o7anHL?'""i?^^^^' '^*'-) ^^r i# 
quantities'ihep?oclltwiarbceoVar "^"^' °^ '^ ^^^^ 

Fo?1:E ^i;;-^ ^£^X^^T'' -'']' - quotient. 
OS 100 kundredhi3'\Z}ihTV^.^i'c,'^'^''^^ ^e considered 

100 ten #/.«W "jM. Sefore t wi ^- "^^ ^' °^^^'« 

will be one /.n S.aS/L ^ Z.^? ' ".T^ "'^ '^'^«"^»<^ 
remainder, must, in the /amP L ^ k ' ^"P *^" thousandth, the 
eth.s; and the nexrquotiTt wIlL "'^^^'f P^^d as ten milHon. 
and so on wi irtl?e oS ni r .""^ '"^'^''''*^'^''«^ -OOOOOI-- 
will be -Ol-i^mi^Som^^^ together, 

by -^01'. ^ ^^wi^^vc, or 010101, &c.--representod 

■ -2^5- (==37XB'fl=37X.^0r) will e-ive -3-37^7 Xr^ 
quotient. Thus ^■o/o/, &c.— or 

010101, &c 
37 






a 3 



70707 
30303 



Tr, +1 .. S73737, &c.=37v-m' 

digits as a period, will bl eauaVto a v ,l ? ^?-"°« ""^ '« 

603 



3003003003 
600600(5006 
5005005005 



T« *!,« ^6^5^3503503, &c=5fi3v\nni 

b >e a cucuiating decimal having these 



CIRCLI.ATING DECIMALS. 



169 



J'pita as a perio(J. — And, consequently, a circulating decimal 
linving any three digits as period will be equal to a vulgar 
truction having the same digits for numerator, and 3 nines 
lor denominator. 

We might, in a similar way, show that any number of digits 
divided by an equal number of nines must give a circulate, 
«ach period of which would consist of those digits. — And, 
Consequently, a circulate whose periods would consist of any 
digits must be equal to a vulgar fraction having one of : s 
j'.eriods for numerator, and a number of nines equal to the 
number of digits in the period, for denominator. 

76. IF tlic decimal is a mixed repetond or a mixed 
nrculate — 

lluLE. — Subtract the finite part from the whole, and 
set down the difference for numerator ; put for deno- 
minator 80 many cyphers as there are digits in the Jinife 
part, and to the left of the cyphers so many nines as 
tliere are digits in the iiijiniie. part. 

ExAfdPLE. — What is \i\Q vulgar fraction equivalent to 
•97^8734' ? 

There are 2 digits in 97, the finite part, and 4 in 8734, 
the intiaite part. Therefore 

978784-97 978G37 . ,^ • , , .... 

~mm~=WMO' '' *^^ '''^1'"''''^ ^^'^Sar fraction. 

77. Reason of the Rule. — If, for example, we multijdy 
•97^8734' by 100, the product is 97 •8734=97 4--8734. Tliis (by 
the last rule) is equal to 97-|-|o|^' which (us Ave multiplied by 
TOO) is one hundred times greater than the original quantity — 
but if we divide it by 100 we obtain TVo+iT-sMfis-'fn. "wlucli is 
equal tjio original quantity. To perform the addition of y^,- 
^^^'^ irf sffiW' ^e must [19 and 22] reduce them to a common 
denomin?».tor — when they become 

97X991*^00, 878400 97X9999, 8734 



99990000 ' 99990000"" 999900 
97x10000-1 , 8734 



10000-1) 
970000—97 



' 999900 
97x10000-97 



8784 
"999900"^ 



999900 ^999900 ~ 999900 

8734 978734-97 978637 . 

999900 "^999900~ 899900 ""999900' "^^^" ^^ exactly the 
result obtained by the rule. The same reasoning would hold 
with any other example. 

EXERCISES. 

7. 



1. 

2. 
3. 
4. 
5. 
6. 



•^8'=5L 

• 73'=T^- 

_14 5 
^057'=rg?X 



^145'= 

^057'= 



8. 

9. 
10. 
11. 
12. 



./)74' 574 

uu -.u — onno- 

•147^658'=:iil^ii 

875-49vG5'=875*^^V' 
301-8275G'=:301^;^^a. 



170 



^•^KCULATINO DECIMAL,. 



vaient vulgar fractio, '^d "° *''"'"''^ "> "««• o„"i. 
''«'«, &0, liie other <le",l!" '""■^ ?''''' -"id subtract 
put Joivn so jaanv of *i "'"'«'/ takin- ca.rtl 

acourac,. """^ »' 'l-eu, as will ^secure "suffieio,;: 

dered as :^2i£2«;s\ .„ *• ^tus a (const 

5 >'«'^^g-™o.act,„otiont;soalso 
W'U i (considered as ™i£2*?x , 

2 ) i^ni \ will not siyo 
»»; fe4 (considered as'-^MSSl^, „, IMJmndn^itl.s 

^ "■■.•'^nkr reason 4 «/,„ 

'-"i->oo4 (considered asl°i-i!s;. 400hXdS: 

%l ''rj„1(';' ^■''.'"Ij contain 7^ ^^ '' 

»« will be eqtl to™e Zlt™ '""^y depi,nal places 
eontamed as factor i„^S 1' "™?''°'- "f twos, or fives 

therefore i2j£f^ " ^^ (^XS; ; and 

'-^^ place 'see. ^7^1^^""" ''^' ^"^ ''' 
quoaeoi. ^^- ^^J), that n, ,,,, ^.^.^^j ^^ 

> 2X2/ ^^" gJve two decimal places • ho. 



CIRCULATING DECIMALS. 



171 



'■gJ, it is not 
tlieir ofjui- 
nd subtract 
"ig cava to 
■e sufficient 

?ar fraction 
' its Jowost 
ctors (fac- ' 
ictors, can 
or noitlior 
tliese — as 
oir inulti- 
ictly con- 
I (eonsi- 

* ', so also 

' noi givo 
idredths. 



ct quo 
ii'Gdtlis, 



places 
1* fives, 
riginal 

'found 
; and 

in the 

al as 

5auso 

■itor, 

•s so 



30 tenths 
niany tenths ; for — ^ (=4 ) cannot give an exact 

quotient — 30 being equal to 3X2X5, which contains 2, 
but not 2X2. It will, however, be sufficient to reduce 

, , , , 300 hundredths 

the numerator to hundredths : because -. 

' 4 

will give an exact quotient — for 300 is equal to 3 X 2 X 

2X^X5, and consequently contains 2X2. But 300 

hundredths divided by an integer will give hmulredths — 

or two decimals as quotient. Hence, when there are two 

twos found as factors in the denominator of the vulgar 

fraction, there are also lioo decimal places in the quotient. 

4*V \r^ o \^ o sy o sy r j contains 2 repeated three times 
^ ,«X'*X'^X<J 

as a factor, in its denominator, and will give three 
decimal places. For though ]0 tenths — and therefore 
6X10 tenths — contains 5, one of the factors of 40, \\ 
does not contain 2X2X2, the othr;> ; consequentlj' 
it will not give an exact quotient. — Nor, for the same 
reason, will 6X100 hundredths. 6X1000 thousandths) 

6 X 1000 thousandths 
will give one — that is, j^ (=4V) ^'"^ 

leave no remainder ; for 6 X 1000 (=6 X2X2X2X5X 
5X5) contains 2X2X2X5. But 6X1 000 Ihausandths 
divided by an integer will give thousandlhs — or threa 
decimals as quotient. Hence, when there are three twos 
found as factors in the denominator of the vulgar frac- 
tion, there are also three decimal places in the quotient. 

81. Were the Jives to constitute the larger number of 
factors — as, for instance, in /^ jf ^j-, &c., the same reason 
ing would show that the number of decimal places would 
be equal to the number of fives. 

It might also be proved, in the same way, that were 
the greatest number of twos or fives, in the denominator 
of the vulgar fraction, any otlber than one of those num- 
bers given above, there would be an eqaal number of 
decimal places in the quotient. 

82. A pure circulate will have so many digits in its 
period as will be e(pial to the least number of nines, which 
would represent a quantity measured by the donoraina- 



172 



CIRCULATING DECiaiALS. 



bo equal to a fraction ),,„/„ J """ '"<"' " e'reulate will 

that IS, It will be eaual +n =. ^ • ^^^ ^^enomiuator— 
of which (the ierlToff}T' ^T'T^ '^'^ numerator 
ii^ the numemtor of the ^,-''''"^' '^ ^^^ ^' «^ ^'^^J 
quantity represen ^d VlL^ST^^r^^^^?^^^ ^ ^^^ 
l^r if a fraction having a^i^'nl '^- '*' ^^"^""nator. 
another which has a alpf f ^ t""'"""'"^^^ ^« ^''l^^l to 
of the latter is to the slm^'. '" ^'T''' ^^'' numerator 

the former-^in which ciTtho""?" '?'' '^^'" *^^^* ^^ 
nierator counteract ttrefe^^^^^^^^ ''^'^ the nu- 

denominator. Thus A-Vf , *^'^ increased sue of tlio 
«f H is 5 times Teaterd^ ' i'TT^ " ^^^' numoratur 

«nd -384615'=5 8 4 el?"! J ^V ^^"^e /^=.-v3846ir/ • 

and, thereforp Jk" V^ '^' ' A' '^^^°' ^^ ^^nai to If a# i5 . 
_ u, tuertiore, whatever mult nIp '^ft.iRi- • »»y*nf'" > - 

J:8 the same of 13 —But qq S • ^f^^'' '^ ^^ ^> ^^''^'^^-^ 
13, consisting of nin^s T? . ' f /^^' ^'''' "^"^^^P^^^ -*' 
Then take f?r numel-ato" sn .1 ' "' T'? "'^^^ ^' ^^^««- 
lesser number of ^ n'; Vs ^^ , i^ '"'"^y' ^^ '^^ ^« that 

number of nines for its deomulltr'Tl.^'' that lesser 
this new fraction will fir^l ^'^l'^'' -^.ho numerator of 

equal to the origina fia^i^on "\ ^ ^''''?^ '^ ^ ^^''^''^^'^tc 
different from 3846 5 Tf '^- ^"^ ^^t^"^ ^«^ Period k 

circulate; there TrCefe'tr^r^^"' *^? ^'^"--' 
equal to r^^-that is two rii!' T t^'"'"* circulates 
for the same frac ioni-wh.V f "'''' ''^^''^'^ ^' --'otiont, 
is absurd to suppo eVa^^a^.v /' '""^'f^^'- ^ 'nee it 
multiple of 13. ^ "^ ^"^ ^"»^^«^ of nines is a 

rarTwh'a^ nSf 2^t1"'fo'"^ ^^^ ?^^^^- '^^ ^-te' 
of the vulgar fraction^-oduced tot ? '^! ^^^^^ominator 
For f7fil n «».•* ^^uucea to its lowest terms 

hand"i/'ti; 't??n' r\' "''• "^P''^- *»"- ■■%>" 

fi-tion,oI,tainXmL 'S,.e:?r,rt; "' f''" "''^'"• 
suppose the (ienominator of t„ ^^»' ';.VP'""-s would 

-tain two, 0,. fivci^i:^ t sniiSrt;:: 



CIRCULATING DKCIMALS. 



173 



to its Icwost 
3irculate wiJl 
01' its nume- 
iiomiuator — 
e numerator 
be as maivy 
tion, as the 
snouiiuator. 
is e(|ual to 
■ numerator 
'an that of 
of the nu. 
' «ize of tlio 
numcratur 
momiaator 

2. 

'3846 If/ ; 

0.4. 0L5 . 
a u ? ft ft ) — 

^, U9!i't>J.4 
lultiple of 
'r bo loss. 
5, as that 
i«it les'ser 
^orator of 
circulate 
period is 
culate of 
Le former 
•irculaios 
'"'otients 
■incc it 
nes is a 

a finite 
>minator 

IS. 

he riojht 

vulgar 

'S would 

tion to 

facturn 



could give cypliers in thnir multiple- 
of the vulgar fraction obtained from the 
If there is a finite 



the dimominat\7r 
ciiculate. 

84. If there is a finite part in the decimal, it will 
contain as many digits as there are units in the greatest 
number of twos or fives found in the denominator of tho 
original vulgar fraction, reduced to its lowest term.s. 

' Let the original fraction be -/-g. Since 56.=J2X2X 
2X7, the equivalent fraction must have as many nines as 
^vill just contain the 7 (cyphers would not muse a number 
of nines to be a multiple of 7), multiplied by as many 
tens as form a ])roduct which will just contain the twos a8 
factors. But we have seen [80] that one ten (which adds 
one cypher to the nines) contains one two^ or five ; that 
the product of two tens (which add two cyphers to the 
nines), contains the product of two twos oi fives ; that 
the pror'.uct of three tens (which add three cyp-hers to tho 
nines), contains the product of three twos or fives, &c. 
That is, there will be so many cyphers in the denomi- 
nator as will bo equal to the greatest number of twos or 
fives, found among the factors in the denominator of tho 
original vulgar fraction. 

]3ut as the digits of the finite part of the decimal add 
an equal number of cyphers to the denominator of the 
new vulgar fraction [7GJ, the cyphers in the denominator, 
on the other hand, evidently suppose an e(|ual number of 
places in the finite part of a circulate : — there will there- 
fore be in the finite part of a circulate so many digits 
as will be equal to the greatest number of twos or fives 
found among the factors in the denominator of a vulgar 
fraction containing, also, other factors than 2 or 5. 

85. It follows from what has been said, that there is no 
number which is not exactly contained in some quantity 
expressed by one or more nines, or b;7 one or more nines 
followed by cyphers, or by unity followed by cyphers. 



Contractions in MtrLTiPLicATiON and Division 
(derived from the properties of fractions.) 

86. To multiply any number by 5 — 
IIulb:. — Remove it one place to the left hand, and 
divide the result by 2 




174 



<'"\ri:.\'TroN'<<. 



KxAMrr.r;, . 7:!Ox/5~."^^h,^;..;^,, 

^'- 1<» imiltipiv \,y 25 _ 3 

divide liM^'""'"' "" 'i'"^»t^'y two places to tho left, and 

M\.\.AnM,K.— 0732x25.^ ''f.iaon..,, .«.,^n 

-^— , ; flH^rcfurc G/.!i»xi:5=G732x'"" 
*^- 'i'o multiply l,y 125 ._ " * 

di^:'^';::::;:i;'i ^^^^-^^^ ^j-- pi^^ec. to the loa. and 

liiOAsox. — 12-5— ''^»"- t) r Tr-^r^-^-^^- 

KVAMl'LE G85 XTSsTT*"' 5"" ^T .,n . - 

lOOx,/. * --I'JUX, : thei-fifore 085x75 = C85x 

'H). To imiltiply by 35— 

left STi!;- l^^'t'S^tfo""?"!- ^"^ I'^-- *« tho 
I'luce to the left. ^ ' ^ *''' "Hiltiphcand removed one 

'f~-fi'0. ' therefore G / 80Gx 35 =67806 x 

p!; J'' t'^' '7'^"^ <^'^'^'>^ the multipliers-- 

Fv^;7 v^ -^ '^^ '"' "^'"^'^^^"^ ^'^^tion. inverted 
^L^.,...4.v.de 847 ,^ 5. 847..5=847-^ V^47x 

easy to divide, a. ro v^ultnAl'C"'^j'-,r^^". '^ ^'."^'^ ^^ 
liUAed number. " ^ "^ -^ •* r ^^> i^« crpnvalene 



DECIMALS. 



m 



^=«^v"==:5'Vfto. 



tlio left, and 



o V ' " " 



QUESTIONS FOR THE PUIMf, 



1 Show tliat a decimal fraclion, and the 



spond- 



^ X ' 



the loft, and 



'SG5='7o. 



"•f^ loft, then 
hy -1. 

5. 

C75 = C85x 



«cns to tlio 
Jinovcd one 

== 109740Q 
= 67896 X 

ornsclves to 



ertod. 
y'=847x 

■ide hy the 
1 wJieiii we 

3h'er v'ill 
is not so 
riuivalciit 



Iccmif 
*"6 "'-•'^"""'i «»>iu iiuu identical L^jyi. 

2. How is a decimal chann;cd into a decimal frac 
iion? [GIJ. 

3. Are the methods of adding, &c., vulgar and deci- 
*nal fractions different ? [62]. 

4. How is a vulgar reduced to a decimal fraction ? 
[63]. 

5. How is a decimal reduced to a lower denomina- 
tion .? [64]. 

6. How are pounds, shillings, and pence changed, ai 
once^ into the corresponding decimal of a pound r [66, 
67, and 68]. 

7. How is the decimal of a pound changed, at once^ 
into shillings, pence, &c. } [70] . 

8. What are terminate and circulating decimals } 
[71]. 

9. What are a repctcnd and a perio Meal, a puro 
and a mixed circulate ? [72] . 

10. Why cannot the number of digits in a neriod bo 
equal to the number of units contained in the divisor } 
[73]. 

1 1 . How is a pure circulate or pure repetend changed 
into an equivalent vulgar fraction ? [74] . 

12. How is a mixed repetend or mixed circulate 
reduced to an equivalent vulgar fraction r [76] . 

13. What kind of vulgar fraction can produce no 
equivalent finite decimal ? [79] . 

14. What number of decimal places must necessarily 
be found in a finite decimal .? [80] . 

15. How many digits must be found in the periods 
of a pure circulate ; [82] . 

16. When is no finite part found in a repetend, or 
circulate ? [S3] . 

17. How many digits must be found in tho finite part 
of a mixed circulate t [84] . 

18. On what principal can we use the properties of 
fractions as a means of abbreviating the processus of 
multiplication and division ? [86, &c.] 






til l<! 



176 



SECTIOJV V. 

PROPORTION. 

numbers are given a W]! ^ •T''"' '^ ^^' ^^^^^^^ tluco 
found. ^ "' * *^"^*^' ^'»«h ^s unknown, may bo 

shown by Hatton, in bis IrthJr i ?^«"''ate, as was 
hundred years ago '^^ '''^^" published nearly one 

p4oS,^::!l tXs:r^i^T;^ ^^^^ f « ^^ 

miportant prf clnles nnnn '1/ !? •?, , '^ ^''^^ ^^^^P^e but 
i he following tmth3 aro self-evident •_ 

quantity, 4 for instance Ix^ Id '• vm'^^ ."'"J'""" 
equal, we shall have 5X6+4=3X10+^ ' """^ ''™ 

equal ^':^l^^.:p:''^ij: ^ «"^" - 

^. 11 the same, or erjinl r<»..^*-*- 
from others whieh a e Tual ^ ''" ''^ '''^''''''^' 
equal. Thus, if we subtCt'- f." ''"^'^f^^^ ^^^ bo 
quantities 7, a'nd S+Cweiu l'.? ""' '' ^'^ ^^^"^^ 
7-3=5+2-3. 

And since 8=6+2, and 4=3 + 1. 

8-4=0+2 -3+T 



PROPORTION. 



177 



tho golden 
' it is termed 
. when thrco 
wn, may bo 

' the simple^ 
divided into 
ate, as was 

1 nearly one 

tlio rule of 
simple but 
•0 of ratios. 



'i to 0(j[ual 
d tlio sum,. 
which aro 

kvhich are 



subtracted 
s will bo 
the equal 



same, or 
Thus 



if wo multiply the Oi\\\v^h TH-fl, and 10+1 by 3, 



bhall lia\ 



we 



G+(]V3=:U4rix3. 
And Kiucu 4 + 0— ID, and 3X().~18. 

4+'JX3xG=i;5xl8. 
0. If equal quantities aro divided by tlio same, or by 
• equtil (|uantitit'.s, the quotients will be oqunl. Thus if 
wo divide tlie C(iuals 8 and -1+4 by 2, we shall havo 

8_4+4 

2 T 

And since 20=17 + 3, and 10=:=2xr). 

20_17+8 
10"~~2x5" 

7. Ratio is the relation which exists between two 
quantities, and is expressed liy two dots ( : ) placed be- 
tween them— thus o : 7 (reatl, 5 is to 7) ; which means 
that 5 has a certain relation to 7. The former quantity 
is called tlic onlccedevty and the latter the covscqi/rnl. 

S. If we invert the tci ms of a ratio, we sliall havo 
their inrenc ratio ; thus 7 : 5 is the inverse of 5 : 7. 

9. The relation between two quantities may consist 
in one being greater or less than the otlier — then the 
ratio is termed arithmctlad ; or in one being some mut- 
tipk or part of the other — and then it is geometrical. 

If two quantities arc ecjual, the ratio between tlunu 
is said to bo that of equality ; if they are unequal it is 
a ratio of greater inequaUty when the antecedent is 
greater than the consequent, and of hsscr inequality 
when it is less. 

10. As the cviithmetical ratio between two quantities 
is measured by their difference, so long as this difference 
is not altered, the ratio is unchanged. Thus the ratio 
of 7 : 5 is equal to that 15 : 13— for 2 is, in each case, 
the difference between the antecedent and consequent. 

Hence we may add the same quantity to both tiio 
antecedent and consequent of an arithmetical ratio, or 
may subtract it from them, without changing the ratio. 
Thus 7 : 5, 7 + 3 : 5+3, and 1 ->2 : 5-2, arc equal 
arithmetical ratios. 

Uut we cannot multiply or dih'ide the terms of an arith- 



I 



if 
It 




173 



PnoPORTlON. 



V r 



"■■0 c,,uul ; thus 10 : 5=12 : fi L ' •- ■ a ^u ","'" 

>-»t.o by the «u,uo number mthoutaltoring^I.e .aao 
U'us 7X2 : VX2=7 : 14-bocauso '^^^ 1 

tion formerly mvcMi " Wl.^f f... 1- 1 f^J" q'les- 
2U ?" «^l,{i • ' ,. ^*^'^'^ Inichon ot a pound is 
7i* , — wJiich in rea ity moans " Wl...f .. / /• • 
there between 2\d. and a pound '' or ''WI ? '? '' 
consider 2U.. i w. nnn«\r 1' „„^/ ^V hat must wo 



or. 



If 



consider 9iZ It- ^^ 1"^""« 5" or " What mu: 

Tn fine '?wi;. -Ti '""f '^ ''^ P^^"^^^ «« "'"^y ;" » 
w ' ^* ^^ ^^>^ value of 2J- • 1" 

terms by the same numb rffif 1 9 • IZt t^""' "^ 
ratio a8jf^:«2 or n // ,m, ™ ** ;, ''™ '? 'ho samo 

f:4i^,iii,rjj:;°st'^^^^^ 

•tuuu nmtij rv!»sonta their ratio, aud unity. Thus 



PROPOUTION. 



179 



•n flio liLst example 9 : 000 and ,J^ : 1 arc equal ratios. 
H If, not iioc'jssary that wo hIiouKI bo able to oxpiess by 
int^igors, nor even by a finite decinuil, what part or mul- 
tiplo one of the terms is of the other ; for a geometrical 
ratio may be considered to exist between any two quan- 
tities Thus, if the ratio is 10 : 2, 5 ( V) is the quantity 
by which wo must multiply one term to make it equal 
to the other ; if 1 : 2, it is 05 (^), a fmite decimal : lut 
if 3 : 7, It IS M28571' (^), an infiuitc decimal— in which 
case wo obtani only an approximation to the value of 
the ratio. 13ut though the measure of the ratio is ex- 
pressed by an mjiiiite decimal, when there is no quantity 
which will exacfiy aerve as the multiplier, or divisor of 
ouo quantity so as to make it equal to the other—sinoo 
wo may obtain as near an npproxunation as we please^ 
there is no inconvenience in supposing that any one 
number is some part or multiple of any other ; tl'at is, 
that any number may bo expressed in terms of another— 
or may form one term of a geometrical ratio, unity 
being the other. "^ 

14._ Proportion^ or analogy^ consists in the equality 
of ratios, and is indicated by putting =, or : :, between 
the equal ratios ; thus 5 : 7===Q : 1 1, or 5': 7 : : 9 : 11 (read, 

5 IS to 7 as 9 : 11), means that the two ratios 5 : 7 and 
9:11 arc eijual ; or that 5 bears the same relation to 7 
that 9 does to 1 1 . Sometimes we express the equality 
of more than two ratios ; thus 4 : 8 : : G : 12 : : 18 : 36 
(vcd, 4 is to 8, as 6 is to 12, as 18 is to 30), m'eana 
there is the same relation between 4 and 8, as between 

6 and 12 ; and between 18 and 36, as between either 4 
and 8, or 6 and 12~it follows that 4 : 8 : : 18 : 36— for 
two ratios which are equal to the same, arc equal t' 
each other. When the equal ratios- are arithmetical, the 
constitute an arithiMtiail proportion ; when geometri 
cal, a geometrical proportion 

15 The quantities which form the proportion are 
called proportionah ; and a quantity that, along with 
three others, constitutes a proportion, is called a f&iLrth 
proportional to those others. In a proportion, the two 
outside terms are called the extremes^ and the two middle 
terms the means ; thus in 5 : 6 : :7 : S, 5 and 8 are tho 



m 



J- \ ■ A 



%• 



180 



PKOPORTION. 



extromos, 6 and 7 the meanf.. mien tlie same qiiantitr 
IS found m bolA means, it is called l/ie mean of the 
extremes ; thus, since 5 : 6 : : 6 : 7, 6 is tAc mean of 5 and 
7. VV hen the proportion is arithmetical, t/ie mean of 
two quantities is called their arithmetical mean • -when 
the proportion is geometrical, it is termed their' ^e^Ts^- 
onml mean. Thus 7 is the arithmetical mean of 4 
and 10; for, since 7-4=10-7, 4: 7: :7:10. i\nd8ia 

the geometrical mean of 2 and 32 : for, since 5 j 

2 : 8: :8 : 32. > ^^^^ s— ai* 

16. In an arithmetical proportion, " the sum of -the 
means is equal to the sum of the extremes." Thus, since 
11:9:: 17 : }5 is an arithmetical proportion, 11-9^:=^ 
17-10 ; but, adding 9 to both the equal quantities, we 
have 11 9 + 9=17-15 + 9 [3]; and, adding 15 To 

n'^'^To h^-' •^^-^+^+^'^=1^-1^+^+15 ; but 

H g^.9 + 1^ jge(^^j.j| ^^ ii + i5_sinco 9 to be sub- 
tracted and 9 to bo added =0 ; and 17-15 + 9 + i5-_ 
17+9_since 15 to be subtracted and 15 to be added =0 • 
therefore 11 + T5 (the sum of the extremes) =17+9 
(the sum of the mean ,.— The same thing mi^ht be 
proved from any other arithmetical proportion^ and, ■ 
therefore, it is true in every case. ■ l\ 

17. This equation (as it is called), or the cqualitv which 
exists between the sum of the means and the sum of the 
extremes, is the te^t of an arithmetical proportion :— that 
IS, It shows us_ whether, or not, four given quantities 
corstituto an arithmetical propor-tion. It also enables us 
to hnd a fourth arithmetical proportional to three given 
numbers— since any mcc^n is evidently the difference 
between the sum of tlie extremes and the other mean • 
and any extreme,' the difference between the sum of the 
means and the other extreme 

A fiT '^r^ •-'^r\L? • 11,^^ *^' arithmetical proportioif,9 
V ;,+- ^1^^ ' ^°^^' subtracting 4 from the equals, 
wo have 1 1 ,one of the extremes) =7+8-4 (the sum of 
the means, mmus the other extreme) ; and, subtracting 7 
we have 4+11-7 (the sum of the extremes minus !no 
of the means) =8 (the other mean). V.'e might in the 
^amu way nna the remaining extreme, or the remaining 
mean. Any othtr arithmetical proportion would hav6 



PROPORTION. 



18i 



answered just as well — hence what we have said is true 
in all cases. 

18. Example. — Find a foiirfa proportional to 7, 8, 5. 

Making the required number one of the extremea, and 

Cutting the note of interrogation in the plac"e of it, we have 
: 8 : : 5 : '? ; then 7 : 8 : : 5 : 8-}-5-7 (the sum of the means 
minus the given extreme, =6) ] and the proportion com- 
pleted will be 

7 : 8 :: 5 : 6. 

Making the required number one of the means, we shall 
have 7 : 8 : : '? : 5, then 7:8:: 7+5-8 (the sum of the 
extremes minus the given mean, =4) : 5 ; and the proportion 
completed will be 

7 : 8 : : 4 : 5. 

As the sum ov the means will be found equal to the sum 
of tlio extremes, we have, in each case, completed the pro- 
portion. 

19. The arithmetlad mean of two quantities is half 
t\\Q sum of tho extremes. ¥oy the sum of the means is 
equal to the sura of the extremes ; or — since tho means 
are equal — twice one of the moans is equal to the sum 
of tho extremes ; consequently, half tho sum of the 
means — or one of them, will be equal to half the sum of 
the extremes. Thus the arithmetical mean of 19 and 

(=23) ; and the proportion completed is 



27 is 



2 
19 : 23 :: 23 : 27, for 19 + 2^=234-23. 

20. If v/ith any four quantities the sum of the means 
is equal to the sum of the extremes, these quantities aro 
in arithmetical proportion. Let tlio quantities bo 

8 7 5. 

As the sum of the means Ls equal to tho sum of tho 
extremes 

8 + 5 = 0+7. 

Subtracting 6 from each of the equal quantities, wo 
have B+fv— 6 = 6 + 7— 6 ; and subtracting 5 from each 
of these, we have 8 + 5-6--5=6 + 7-6— 5. But 
8 + 5=-6 — 5 is equal to .R — 6, since 5 to be added 
and 5 to bo subtracted are ?=;0 ; and +6 + 7—6—0 = 
7—5, since 6 to be added an^l 6 to '/)Q subtracted =0 ; 

I 2 



.82 



PROPORTION, 



V' m 



tlierefore 8+5 — 6 5 R4-7 p, r. • .l 

fortion. It might in the same way be n-oyed Ui»i 
«y e^te ibnr quantities are in arithnfeacal^p^I^orUon 

21. In A gamttrical proportion, "the jiroduct nf 

t^-'J fulfil ''' ^6* 8 IS a geometrical proportion, 
L~hv 7 2\ ' ^u tiplymg each of the equal quantl- 
nes by 7, we have (V»X7^ — 'Jv7. onri ^ u- i • 

.aehofthe.eby8,weUelfef6x'7(ox7''P'r^ 
•ut 14X8 13 the product of the extremes -and 1 6^7 
B he product of the means. The same"eLonTn« lull 

r" rs^\ri;ter«-' '''»^-'-> -^^- 

Mt^a^-r^rpfet^^^^^^^^ 

7X^S2-lVxnV iV f •^.- *''^g^°»«*™'J proportion, 
/ X ^—HX n i and, dividing the equals by 7, we hayo 

32 (one of the extremes) =1*^ (the product of the 
mo^ns divided by^the other extreme) ; and, dividing these 
by ll,wehaye-jj-(the product of the extremes di- 
vided by one mean)=14 (the other mean). We miBht 

l^l^Zli t 'l^' proportion would have answered 
just as well — and thfirfifm-o wKo* l__. _ -i . : 

in every case. ^ """" "'' "*'" ^^"^ ^ ^^^^ 



PROPORTION. 



ibH 



mo as 
5, are 



itSS. Example. — Find a fourth proportional to 8, 10, and 14. 
Making the required quantity one of the extremes, we shall 

10X14 



Lave 8 : 10 : : 14 : ? ; and 8 : 10 : : 14 



8 



(the product 



of the means divided by the given extreme, ==17-5). 
And the proportion completed will be 

8 : 10 : : 14 : 17-5. 
Making the required number one of the means, we shall 

8x14 



have 8 : 10 : : ? : 14 J and 8 : 10 



10 



(the product of 



the extremes divided by the given mean, =11-2) : 14. 
And the proportion completed vnll be 
8 : 10 : : 11-2 : 14. 



£X£RCISf:8. 



Find fourth proportionals 



1. To 8, 
6. 



6. 



6, and 12 

8 



6, 12 

10, 150 

1020, 68 

160, 10 

68, 1020 



68 

150 

1020 

10 



Jlns. 24. 



16. 



1020. 

10. 

68. 

160. 



24. If with any four quantities the product of the 
means is equal to the product of the extremes, these 
quantities are in geometrical proportion. Let the 

quantities be 

5 20 6 24, 

As the product of the means is equal to the prod. !t 
of the extremes, 

5x24=20x6. 

5X24 20X6 
Dividing the equals by 24, we have""^^ — = 24 ' J 

5X24 20X6 
and, dividing these by 20, we have 20X24 



But::^ 



5X24 



_5 20X6 

=20 5 and 20x24' 



20X24- 

_6^ 5 _6 ^ 

=24' j therefore 20 ~ 24 * 



20X24 
consequently the geometrical relation between 5 and 20 

two equal geometrical ratios — or a geometrical propor 



184 



PROPORTION. 



the pTodJotTelnZll «°™r'"f proportion, I 
eztromes. °^ '^ '^l""' '» ">« P'-oduot of the 

proportljfnal^""' '^'' '^™ '^ ™"y. '-> «■><• a fourth 

a{i;«._Knd the product of the second and third 
-Example, — Whnf i'<a +1,^ ^ ^ i-uira. 

2^ ^ vvhat IS the fourth proportional to 1, 12, and 

W ^ • ^^ • •' 27 : 12x27=324 

«nee dividing a nui fy "^u/dt^Tot'il?" u'^ '"^^- 



EXKRCISES. 

Find fourth proportionals 



,X-Tol, 17, and 8 

}?• » J' 23 „ 20 

J|- » J. 53 „ 110 

^*»- •• I, 15 .. 1234 



^n*, 



j» 



136. 

460. 

7300. 

6830. 

18510. 



by the first. ' ""^ "^ *''^°' ''iM' is not unity 

Ex.«P„.-Find a fourth prop„ra„„„ to 8, 1, „ud 5. 

4-"^ut'it':ftL^;:etilV, "--."yt"" given 
product of both, ™hen tl'e otC hZu^ "JT^dered ^ tho 
pi-t,on .y „„H, p,„auce. no Sn^V?;- Sr'"" 

EXERCISES. 

Find fourth proportionals. 

fi on „.. 1 



H. To 

15. „ 

16. ,. 
17. 
18. 
19. 
20. 
21. 






5. 

6, 
7, 
8, 
6, 

37, 



^?00, 1000 
200, 1 



20, and 

1 
21 
24 

1 

1 






1 

20 

1 

1 

50 

68 

1 

1000 



^?i 



4 

4. 

3. 

8. 

Si. 

4. 

6. 

6. 



= ' ^--"-^^or tiio extremes; and the proan^" 



ROLE OF PROPORTION. 



185 



of the extremes is equal to the mean multiplied by itself. 
Hence, to discover the geoinclriccl mexin of two quan- 
tities, we have only to find some number which, multi- 
plied by itself, will be equal to their product — that is, 
to find, what we shall terra hereafter, the square root 
of their product. Jhus 6 is the geometrical mean of 3 
and 12; for 6X6=3X12. And 3 : 6 : : 6 : 12. 

28. It will be useful to make the pupil acquainted with 
the following properties of a geometrical proportion — 

We may consider the same quantity either as a mean, 
or an extreme. Thus, if 5 : 10 : : 15 : 30 be a geometrical 
proportion, so also will 10 : 5 : : 30 : 15 ; for we obtain the 
same equal products in both cases — in the former, 5X 
30=10 X 15 ; and in the latter, 10 X 15=5X30— which 
are the same thing. This change in the proportion ia 
called inversion. 

29. The product of the means will continue equal to 
the product of the extremes— or, in other words, the 
proportion will remain unchanged — 

If we alternate the terms ; that is, if we say, " the 
first is to the third, as the second is to the fourth" — 

If we " mnltiplijy or dimk the first and second, oi 
the first and third terms, by the same quantity"— 
If we " read the proportion badcwards''^ — 
If we say " the first term plus the second is to the 
second, as the third plus the fourth is to the fourth"— 
If we say " the first term plus the second is to the 
fii'st, as the third plus the fourth is to the thii-d"— &c. 



RULE OF SIMPLE PROPORTT.ON. 

3D. This rule, as we have sr.id, enables us, when threa 
quantities are given, to find a fourth proportional. 

The only difficulty consists in stating the question ; 
when this is done, the required term is easily found. _ 

In tlie rule of simple proportion, two ratios arc given, 
the one perfect, and the ether imperfect. 

31. IluLE— I. Put '^^hut given quantity which belongs 
to the imperfect ratio in .u third place. 

II. If it appears from the nature of iho- question that 
the required quantity must bo greater than the other, 



J86 



RULE OF PROPORTION. 



■!!m ! 




4%o~};^aY^^Tir^-^^^^^ 7-^3 fa wall in one 

It will faoiKrt Jb aLi Tth« n 'm *^^'T' ""^'^ ' 

question briefly, as fo lows-uini JZF^}^- ^^^^ ^^^'^ *''« 

represent the required qlLti^yl^ "^ ''"*' '^ interrogation to 

5 men. 
10 yards. 
21 men. 

'Jyai'da. 

^r^^ps!z a;'iat ^■"'^^^^^^^ '^*^-^^ --^. 

P-/Jratio7an^d\^S3^,r"^*^*^^^ ^^»«h form the 
than 5 men the VpnmW "'i*^ ^ §'^'^*«'' ""'"^^r of yards 

than the^";, nu Jber~h«^^^^^^^ ^^'^' ^^" ^« S'-^ater 

term of the peSt ra^n ^n Ji,'° **"'' ^^ ^' ^° P"<= the larger 
the first plac?!I '^ '"^ *^° «^'°°'^' «^d *he smallerlu 

A ^ 5 ; 21 ; : 10 : ? 

And, completing the proportion, 

5 : 21 : ; 10 • ^1 X 10 .o ^u . 

■ ~~5 =^A *he required number. 

last 5 .en . m: tSt'^:^'Z,^g ?„/■- -8". i. to 

3 men. 
2 days. 
5 men. 
*? days. 

tiitrbV^u^fiTher^ii'"^"^"' ""«'-^« ■»-'' 

„„ "'t^.'rs?,'' 'h" nnmbor of men, the shorter «.„ ,;„» „ .:..._ 
4««nt..j, „, .,re,ui Will last thorn; but th«i',»7,^ «— g'Ji^ 



RULE OF PROPORTION. 



187 



required quantity — henco, in t'his case, the greater term of 
the perfect ratio is to bo put iu the first, tmd the smaller in 
Uio second place— 

5 :3::2:'? 

And, completing the proportion, 

-=1|, tlie required term. 



5:3::2 



5 



34. Example 3. — If 25 tons of coal cost £21, what will 
be the price of 1 ton '? 



25 : 1 : : 21 



pounds £.jp=lG5. 9-^-f/. 



25 



25" 



It is necessary in this case to reduce the pounds to lower 
denominations, in order to divide them by 25 ; this causea 
the answer, also, to be of different denominations. 

35. Rkason of I. — It is convenient to make the required 
quantity the fourth term of the proportion — tliat is, one of the 
extremes. It could, however, be found eqxially well, if conHi- 
dered as a mean [23]. 

Urahon of II. — It is also convenient to make quantities of 
the same kind the terms of the game ratio ; because, for in- 
Btancc, wo can compare men with men, and days with days — > 
but wo cannot compare 7ne7>. with days. Still thero is nothing 
inaccurate in comparing the number of one, witli the number of 
the other ; nor in comparing the number of men with the quan- 
tity of work they perform, or with the nximbtr of loaves they 
eat ; for these things are proportioned to each otlier. Hence wo 
shall obtain the same result whether we state example 2, thus 

6 : 3 :: 2 : ? 

or thus 5 : 2 : : 3 : ? 

When diminishing the kind of quantity which is in the per- 
fect ratio increases that kind whicli is in the imperfect — or the 
reverse — the question is sometimes said to belong to tlie inverse 
rule of three ; and different methods are given for the solution 
of the two species of questions. But liatton, in his Aritli- 
mctic, (third edition, London, 1753,) suggests the above gene- 
ral mode of solution. It is not accurate to say " the inverse 
rule of three" or " inverse rule of proportion ;" since, although 
there is an inverse ratio, there is no inverse proportion'. 

Reasoiv of III. — We multiply the second and third terms, 
and divide their product by the first, for reasons already given 
[22]. 

The answer is of the same kind as the third term, since 

iicitiici' tiic iiiullipliuukiuu, iiur aixxi UiTinluii Oi inia ici lii ixixa 

changed its nature ; — 20*. the payment of 5 days divided by 6 



188 



RULE OF PROPORTION. 



Of J da, ,„„u,p,iea b, 9 givo» ^- x as the pa,„o„t of u 

would not to the 4u^;a'';„;X;i^« «;»;,-<' tl,are.ore ij 

scc'oL,^«"fiLfa„°d'','i,i?,T"™\'" 'i'""^" *'"> «'■»' ""•' 

mo„ m'ca ure?wbe\ tW o""'' ''^ ""='■• ^■"■•"'=«' <«"■'- 
<itlier [29J. ^ '"'"" "'o ""'"poato to each 

Ex.„P..._,f 30 cwt cost ^24, what «„ 27 owt. co»t ' 
Dividing the first and second by d we have 
And, dividing the first and'third" by 4, 

EXERCISES FOR THE PUPIL. 

^ ^^"J a fourth proportionalto 

4. 6 yard, : I yard : : 27,, Am. 4,. Cul 

b. 5 lb 1 ib : : 155. ^,„ g^ 

7. 4 yard, : iSyards : : u. Am. 4s. 6d. 

J? ^t Ssf *»^^^^^^ii^^^o to at £25 p„r 
p!ooe/eo:"?'Z."'^?!;;S "°^' ^2^. '«w much will 50 

Ans. 121 „,"„(?„ ' '""° "■""" "'«y »"ffi» for 32 r 

■ cwt"; 'aT. JIH^' '-'■ °f "-^J" -.St at 50. PC, 

«.o latter shall I ro.;iror''l:"2o";ardV'"" "'""'' "' 



the paymont 
payment of U 

ns the third, 
therefore it 

lio first and 
iatosfc coni- 
ite to each 



IS. £270 



£23 per 

I will 50 

i last 40 
for 32 : 



50.y. 



poj 



h wide, 
iiucli of 






RULE OF PROPOUTION. 



189 



, ,13. At 10.?. per barrel, what will be tho price of 130 
barrels of barley ? Ans. £Q5. 

14. At 5s. per lb, what will be the price of 150 ft) of 
tea ? Ans. 7505. 

15. A merchant agreed with a carrier to bring 12 
cwt. of goods 70 miles for 13 crowns, but his waggon 
being heavily laden, he was obliged to unload 2 cwt. ; 
how far should he carry the remainder for the same 
money ? Ans. 84 miles. 

lo. What will 150 cwt. of butter cost at £3 per cwt, } 
Ans. £450. 

17. If I lend a person ^£400 for 7 months, how much 
ought he to lend me for 12 > Ans. £233 6s. 8d. 

18. How much will a person walk in 70 days at tho 
rate of 30 miles per day .? Aiis. 2100. 

19. If I spend £4 in one week, how much will I 
spend in 52 ? Ans. iS20S. 

20. There are provisions in a town sufficient to sup- 
port 4000 soldiers for 3 months, how many must bo 
sent away to make them last 8 months ? Ans. 2500. 

21. What is the rent of 167 acres at £2 per acre ? 
Ans. £334. 

22. If a person travellmg 13 hours per day would 
finish a journey in 8 days, in what time will he accomplish 
it at the rate of 15 hours per day > Ans. 6|f days. 

23. What is the cost of 256 gallons of brandy at 12s. 
per gallon ? Ans. 3072s. 

24. What will 156 yards of cloth come to, at £2 per 
yard .? Ans. £312. 

25. If one pound of sugar cost 8^Z., what will 112 
pounds come to .? Ans. 896d. 

2b. If 136 masons can build a fort in 28 days, how 
many men would be required to finish it in 8 days } 
Ans. 476. 

27. If one yard of calico cost 6^., what will 56 yards 
come to ? Ans. 33bd. 

28. What will be the price of 256 yards of tape ak 
2d. per yard ? Ans. 612d. 

29. If £100 produces me £6 interest in 365 days, 
what would bring the same amount in 30 da^'S .'' Ans 
j&i:<5iD iJi. 4a. 






'♦i 



,f 






i 






m 



14*0 



nuLE OK I'noromio.v. 




30. What shall I receive for 157 pair of gloves, at 
I Of/, per pair? Ans. 157 Od. fa > «*" 

31 What would 29 pair of shoes como to, at 9* ner 
pair? Ans.2Gls. v., »t ^j. per 

aJ^: /- "^ ^TT ^'°^, ^"' neighbour a cart horse which 
draws lo cwt. for 30 days, how long should he have a • 
horse m return which draws 20 cwt'? Ans. 22i lys 

rr\l%a- "''"' P'\*^ ["^ '"*«'''''^ ^<^ -^'^ V^r cent." would 
give £6 m one month ? Ans. ^£1200 

34 lfllendi2400for 12 months, how lone; our^htJEl. 50 
be lent to me, to return the kindness ? Am. 32 months 

hsflO^orS' 'V ^^^^"^^"/'^re found sufficient to 
last 10,000 soldiers for 6 months, but it is resolved to 
add as many men as would cause them to be consu.hcd 

Anl 20 000. ' ''""'^'' '^ ™'" '""''^ '^'^ ^^"^ ^^ ^ 

/•n.^o ^^ ^,,^°7<^''' subsist on a certain quantity of hay 
for 2 months, how long will it last 12 horses ? A^^ 
1} months. 

n/i^'/ '^^^P^^f'eper is so dislionest as to use a woi<rht 
of U for one of 16 pz. ; bow many pounds of just v 
be equal to 120 of unjust weight ? Ans. 105 lb 

rlnw • r' ""^ ^^^ **" ^^ '"^^^^^ ^J 40 men in 10 
Ans.Vs^ days.""''^^ ""'^^^ '' ^' ^'''"^''^ ^^ ^^ "^^° •' 

are^Lt^IfTl *''' ^''^^'^ ''-'^"^^ ^'™' ^^^^^^ proportion 
are not of the same denomination ; or one, or both of 
them contain different denominations— 

HuLE.— Eeduce both to the lowest denomination con- 
tamed m either, and then divide the product of the 
second and third by the first term. 

pomrcoTt /-"^^ '^''" '''''''' '^ '''' '''' 1^^^- ^vh'-^t ^ill 87 
The lowest denomination contained in either is ounces. 

':■. ?_ '■ 1302X15 d. 
- ^ • j^J • • 15 : 3— =6960=£29. 

1392 ounces. 
There is evidently the same ratio between 3 oz and 87 Th 
as between 3 oz. and 1392 oz. (the equal of 87 ft) 



RULE OF rnOPORTION. 



101 



ExAMPi.E 2.— If 3 yards of any thing cost 4^. 0J(/., what 
can 1)0 bought for £z i- 

The lowest denomination in either is farthings. 



s. d. 
4 9? 
12 

57 ponce. 
4 



20 231 

40 shillings. 



nls. 
3. 



231 farthings. 480 pence. 



1920 farthings. 
There is evidently the same ratio between 4*. ^Id. and /2, 
fls between the numbers of farthings they contain, respectively 
For there is tlio same ratio between any two quantities, us 
between two others which are equal to them. 

Fa'amplk 3.— If 4 cwt., 3 qrs., 17 lb, cost XIO, how much 
will 7 cwt. 2 qrs. cost ? 

The lowest denomination in either is pounds. 

f' 840x10 

19 : • ^^,^ =£29 Is. bd. 



cwt. 


qr. 


lb 


cwt 


qr. 


4 


3 


17 


: 7 


2 


4 






4 




19 ( 


^rs. 




30 < 


:|r8. 


28 






28 





549 lbs. 



840 K)3. 



EXERCISES. 

Find fourth proportionals to 

39. 1 cwt. : 17 tons : : £5. Ans. £1700. 

40. bs. : £20 : : 1 yard. Ans. 80 yards. 

41. 80 yards : 1 qr. : : 4005. Am. Is. 3d. 

42. 3s. 4d. : £1 10s. : : 1 yard. Atis. 9 yards. 

43. 3 cwt. 2 qrs. : 8 cwt. 1 qr. : : £2. Ans. £4. 

44. 10 acres, 3 roods, 20 perches : 21 acres 3 roods : 
£60. Ans. £120. 

45. 10 tons, 5 cwt., 3 qrs., 14 ft : 20 tons, 11 cwt , 
3 qrs. : : £840. Ans. £1680. 




109 



i.ja* 



RtTLE OK PROPORTION. 



cwf ? ^t' 'lltoo'"^° '' '' "^^ ^' '^"^^' '' "^^ P- 
pricL?i5irr'if ^-*« ''-> -Hat will bo tho 

yar^dt inri ^nn^^/ •Z"*^ '°'*' ^^ ^'- ^^^* ^iU no 
17 .tf o °^*- ^^^^""er costs ^26 6^., how much 4ill 

^7^ 8^1 cwt ^'' '"^"^ ''^''' can I have for £615 isi."? 
57. How much beef can be bought for £760 12* al 

1 ,1 i ^^' ^ ^''•' "^ ^^*-' cost £150, what will 3 ft, 

1 oa., 1 1 dwt., cost ? Ans. £37 105. ' 

69 If 10 yards cost 17.., what will 3 yards 2 ars 
cost? Am. bs. Uid. -» j'»iUH, -* qrs. 

60. If 12 cwt. 22 ib cost £19, what will 2 cwt ^ 
qrs. cost ? Am. £4 5*. 8^^. "^ ^m ^ cwt. 3 

n^nV 14 ^^ ""'•'/? ^7*-' ^^ g"'> «««* 19*-> what will 
13 oz. 14 grs. cost ? Am. 15s. lOd. 

mination- ' ^^''^ *°''"' '°'''''*' °^ "'^''^ *^'"-^' ^'^« ^cno- 

if "^rf •■""■^fi!^"''^ '^*? ,*^^ ^^^^'^* denomination which 
contains then multiply it by the second, and div de 

the produc by the first term.-The answe^ wfll be of 
hat denommation to which the third has been reduced 
u^ rnay sometimes be changed to a higher [Se^ 



RULE OF I'ROPORTION. 



19S 



Example 1.— If 3 yards cost ds. 21(1, what will 327 yards 

Tho lowest denomination in the third terri is farthings. 

yl"- ^i"- *• i\ 3'^7x441 £ s. d. 

3 : 327 ; : 9 2| : ^ farthing8=50 1 6|. 

12 

^)i. 110 pence. 

4 

441 farthings. 

Kx AMPLE 2.— If 2 yards 3 qrs. cost 11 W., what will 27 
yards, 2 qrs., 2 nails, cost 1 

Tho lowest denomination in the first and second is nails, 
and in the third farthings. 



yds. qr. yds. qr. n. 
2 3 : 27 2 2 
4 4 



d. 
lU 



442x45 

— 4^ — farthing8=9<. bd. 



11 qr. 110 qr. 
4 4 

44 nails. 442 nails. 



45 farthings. 



Reducing the third term generally enables us to perform the 
required raultiplicatiou and division witli more facility. —It ia 
sometimes, however, unnecessary. 

Example.— If 3 lb cost £3 lis. 4\d., what will 96 lb cost? 

n> lb £. s. d. ^ s. d. £ s. d. £ s d 

3: 06:: 3 11 4; : ' '^"^ =3 11 4Jx32=114 4 8 



EXERCISES. 



Find fourth proportionals to 

62. 2 tons : 14 tons : : ^228 10*. Ans. 199 10*. 

63. 1 cwt. : 120 cwt. : : 18^. 64. Am. .£111. 

64. 5 barrels : 100 barrels : : 6s. Id. Ans. £6 Us. Sd 
. 65. 112 ft) : 1 ft) ; : ies 10s. Ans. l{d. 

r66. 4 ft) : 112 ft) : : b\d. Ans. \2s. 3d. 

67. 7 cwt., .3 qrs., 11 lb : 172 cwt., 2 qrs., 18 ft) : : £,3 
9s. A\d. Ans. £87 55. Ad. 



n ;""!i 



194 



RULE OF PROPORTION. 



68 172 cwt., 2 qrs., 18 lb : 7 cwt., 3 qrs., 11 lb : : ^87 
6*. 3^^. A71S. £3 195. 4id. 

Am'Jl ^^^•' ^ ^^■^•' I'* * * 2 cwt., 3 qrs., 21 lb : : £73 

70. £87 Gs. 3d. : £3 19s. 4-i^. . : 172 cwt., 2 qrs., 18 
ib. Ans. 7 cwt., 3 qrs., 11 lb. > ^ ' 

71 £3 195. 4irZ. r £87 65. 3r/. : : 7 cwt., 3 qrs., 11 lb. 
Am. 172 cwt., 2 qrs., 18 ib. > ^ » 



/l^^^'^tll^^^" ^'^' ^^"^ ^^*->^^a* ^ill 120 cwt. cost.? 

^.L' £1^05*^4/" ''""'' ^^^''^"^ ''^ ^ ''^' ''' 

74. What will 120 acres of land come to, at 145 6d 
per acre.? ^w5. £87. ' ' 

75._ How much would 324 pieces come to, at 2s S^-d 
per piece ? Ans. £43 175. 6^/. ' f ' 

76. Whafr is the price of 332 yards of cloth, at I65. 
4^/. per yard .? ^7*5 £107 1 65. 

i/^" ■'•^ l?^^^^ ^^ ^^''"^^ ^^^^^ ^'' 4d!-) what will 18 lb 
10 oz cost.? Ans. £49 135. 4d 

t. ?'L'. £1^2 13?4f ' "'^^ "'^ ' ^"^- ^ ^^- -- 

rent ( I 156 acres 3 roods .? Ans. £089 I45. 

^fl" ^^* }^^' ^'^' P^^ ^''•' what will 56 cwt. 2 qrs bo 
worth.? Ans. £118 13.S-. ^ 

81. At 155. 6^ per yard, wliat wHl 76 yards 3 qrs 
come CO .? Ans. £59 95. 7id ^ 

lb ?' 2' £r065.'' """'' ' "^ ''"' *'' ^* ^'' ''• P«^ 
83 At 145. 4d. per cwt., what will be the cost of 12 
cwt. J qrs. .? Ans. £8 195. 2d. 

84. How much will 17 cwt. 2 qrs. come to, at 195. 
lOJ. -^er cwt. A71S. £17 75. Id. 

2 n?; "^i"""*- °^^^"«^«osts £6 65., what will 17 cwt , 
2 qrs , 7 lb, come to .? Ans. £102 125 lOi^^ ' 

■;■ .^IJ ^^- ^'^ ^ cost ^^; lo.v. 9^ , Avhat will be the 
cost of DO cwt., 3 qrs, 24 Jb .? Ans. £378 I65. 8^1 



RULE OF PROPORTION. 



195 



87. If tlic shilling loaf weigh 3 ft 6 oz., when flour 
sells at £1 13s. 6d. per cwt., what should be its weight 
when flour sells at £1 7s. 6d ? Ans. 4 lb 14f oz. 
i,£8. If 100 lb of anything cost .£25 Bs. 3d.,\lmt will 
be the price of 625 lb ? Ans. £WS 4s. 0-^~d. 

S9. If 1 lb of spice cost 105. Sc^., what is half an oz. 
worth > A'ns. Ad. 

90. Bought 3 hhds. of brandy containing, respectively, 
Gl gals., 62 gals., and 62 gals. 2 qts., at Qs. Sd. per 
gallon ; what is their cost.? Ans. £Q1 16s. 8d. 

39. If fractious, or mixed numbers are found in ono 
or more of the terms— • 

KuLE. — Having reduced them to improper fractions, 
if they are complex fractions, compound fractions, or 
mixed numbers— multiply the second and third terms 
together, and divide the product hy the first — according 
to the rules already given [Sec. IV. 36, &c., and 46. 
&C.J for the management of fractions. 

Example.— If 12 men build -3^ yards of wall in ? of a 
week, how long will they require to build 47 yards 1 
Sf yarJs=2,6 yards, therefore 
. . i^X47_, 



26 

1 



47 



7 



=9]- weeks, nearly. 



"0. — If all the terms are fractions — 
lluLE.— Invert the first, and then multiply all the 
terms together. 

ExAMPLK.~If f of a regiment consume \l of 40 tons of 
flour in | of a year, how long will ^- of the same regiment 
tako to consume it ? 

i'l--V- f Xl-T-|=^XfX?=,^=202-8 days. 

Tin's rule follows from that which was given for the division 
of one fractiou by another [See. IV. 49]. 

41. If the first and second, or the first and third 
terms, are fractions-r— 

Kui.E. — llediuie them to a common denominator 
(should they not have ono already), and then omit tho 
denominatorsi 



i ' 



S^K i 



li(f 



\m 



'Ji:li 




RULE or PROPORTION. 

a o^^t'cS"" ' "^ ^ ™'- "^ ""' «™'' •^2, what ,iU ^ of 

I : .J . . 2 : "J 
Reducing the fractions to a common "denominator, we have 
, , . fff •• U :: 2:? 

And omitting the denominator, 

20:27::2:2^=£2-7=£2 14,. 

andVhiiH?®'"^^^^.""".^.*'^^^^"^ *^® fi'-^t and second, or the first 
BO^it^tTopitr"'^^" denominator-whiclfc^S^I Zll 

EXERCISES. 

cos'ti' ^Aii'ijr '"*' '"' "^^' ^"^ ' ^ ^^ -• 

aI%^''' """'^ ""^^ ^ ^"'^ '^^^ *« if 1 «ost is. ? 

of 'S^'^t^^'T^ ^^^^ ^^^-^ ^'^' ^« *^^ !>"- 
^r'^7?f 4T]' '« ^^- «^ «^^-^ -«^ ^' 6i- per oz. .. 

I h'avo V^l^f^'^Tk^f-^ '^" "'^°^ P™^^ -« 

cosf ^7^«f i!^ *^ ?"'^ ^^ ^^T=V yards of cloth, if 7f 
cost i^7 lb5. 4d. .? ^W5. iE51 35. 113 3^ ' • 

R. ^??;7 ^l ^^^^.vi '^^^^ '' ^^'*^ ^981, what will ie363 
85. 7^d. be worth .? ^?w. ^^^358 7, ij' ''^'* 

bought for ^i2'3''pP'l^ for 4| yards, how much can bo 
Dougttt tor £2j\ > Ans. 24 yards, nearly. 

MISCELLANEOUS EXERCISES IN SIMPLE PROPORTION. 

102 Sold 4 hhds. of tobacco at 10ifZ ner TK • INTn 1 
weighed 5 cwt., 2 qrs. ; No, 2- 5 ^™' V •_ P , ^« «- ^ 



iTirf 



1 ^.. 1 /< K 



•»T 



lb ; and No. 4, 5 cwt., 1 



^j'., X i lu : ISO. 



pnoe.? ^7M £]04 Us. 9d 



, 1 qr., 21 lb. What 



RULE or PROPORTION. 



197 



1 03 . Suppose that a bale of merchandise weighs 300 Jb, 
and costs £15 45. 9d. ; that the duty is 2d. per pound ; 
that the freight is 255. ; and that the porterage home 
is Is. 6d. : how much does 1 ib stand me in ? 

£ s. d. 

15 4 9 cost. 

2 10 

1 5 

1 



ft) 

300 



lb 
1 



duty. 
freight, 
6 porterage. 



: 19 

20 

•iOT 

12 



1 3 entire cost. 



300)4575 



15|d. Answer. 

104. Heceived 4 pipes of oil containing 480 gallons 
which cost 55. 5^d. per gallon ; paid for freight 45. pet 
pipe ; for duty, 6d. per gallon ; for porterage, l5. per 
pipe. What did the whole cost ; and what does it stand 
me in per gallon > Ans. It cost £144, or 65. per gallon 

105. Bought three sorts of brandy, and an equal 
quantity of each sort : one sort at 55. ; another at 65. ; 
and the third at 75. What is the cost of the whole — 
one gallon with another ^ Ans. 6s. 

106. Bought three kinds of vinegar, and an equal 
quantity of each kind : one at ^^d. ; another at 4d. ; 
and another at 4Ji. per quart. Having "mixed them 
I wish to know what the mixture cost me per quart } 
Ans. Ad. 

107. Bought 4 kinds of salt, 100 barrels of each ; 
and the prices were 145., I65., 175., and 195. per barrel. 
If I mix them together, what wOl the mixture have cost 
me per barrel } Ans. I6s. 6d. 

108. How many reams of paper at 95. 9<i., and 
125. 3d. per ream shall I have, if I buy £55 worth of 
both, but an equal quantity of each .? An,s. 50 reams 

/\T an on 

109. A vintner paid £171 for three kinds of wine : 
one kind wa,s £8 IO5. ; another £9 55. ; and the third 



'if; 



lUS 



%!' 



RULK OF riioroRTroN. 
Iia.l of 



•fiJO l')s, nor hjii? TL. I,., j r» i 
♦''-'--•'' orchid. ^^^^^^^^^^^^ 



10 15 



28 



10 



2H 10, the prioo (.f throo J.og.shoadH of oa.,h 



£ 

171 



, X171x;{ 
■ £2H 10~^^ ^'''Js. 



I)arr,-I.s had I „f oach > yi,,, yoo ■*-'•"• ^'"'■f '"•■"ly 

weeks. "^^ weoiv ^ Ans. 56 

provisions. Ifow lot' w^l/'^'^?^^^*'' ^^''^««« ^^^ ^>f 
and 2 ^^y^, ^ ^''"S ^'" *''^^3' ^^^ ^ ^ns. 26 weeks 

page. At .'],at |,,e n^I^ Te eVpo'ct'uo ho" '?''^^ 
copy contaming 400 pa/s P ^;.'^; %f;«"j;n « 

il^^^.y^^Z^7 'V^^^ "«'«b^r of 'each: 

117. Suppose that a i^reyhoiiiul molroc 07 
>vlnle a Jmre makes O;! ?nd h 1. • '^ 'P''"'S^ 

rqunl Icii.r(h T„ i„„; ,' ^"'^'^ '''^"' «P'"ings are of 

«^^'t.ikc.i, u .he IS au .prmgs before (he hound ? ' 



RULE or riiorouTiON. 



The tinio tukc^n by tlio gi-c^huuiid for ono 



that i-oquinMl by ihc hnw., ah 2') : 27 



as 1 : n 



iJ9 

Bprin/i; in to 



^^^ U-J- 'I'J'u ^';r('yliomi(l, ilicn'iuio, iraiiiif "'.^p of 



OV U8 



Hpnng Uunug ovcry ^prin;,' of (1h! luirc. 'Jlicrol 



oro 



tl»c 



: 50 : : 1 
liavo will m 



snriii;^; : 5()-^^",=rzG7^), Ihc number of Hprinjra 
ako, bi'l'oro it is overtake:!. 



10 



118.^ If a tun of tallow oo.sts ,£35, nnd iH sold at tl . 
rate of 10 per cent, profit, what in tho solliinr prico > 
Ans. JU3S U)s. ° 

119. If a ton of t;ilIo\v costs ,£.17 10.y., at what rntt 
muMt it bo Hold to gala by U) tons tlio price of I ton > 
Afis. £40. 

120. JJought 45 barrels of boof at 21. v. per barrel; 
auion,!^ tliem aro IG barrels, 4 of whi(3h would bo wortii 
only li of tho rost. Mow muoli must I pay t Ans. 
£43 l.v. ^ -^ 

121. If 840 oggs aro bought at the rate of TO for a 
penny, and 21G more at 8 for a peiniy, do I lose or gain 
if I soil all at ly for 2d. ? Avu. I gain ikl. 

122. Suppose that 4 men do as much work as 5 
women, and that 27 men reap a (juantity of corn in Mi 
days. In how many days would 21 wonum do it .? Ans. 

Tlio work of 4 mcn=that of 5 women. Thorefore (divldinj; 
each of tho equal quantities by 4, they will remain e(pialj^ 
4 men's work . , s <hu work t)l'5 women 
-^ (one mans work )= - ■ ^ . Con- 

bcquently Ij times tho work of one woman=rl man's work .--^ 
that is, tho work of oiio man, in t(U'ma of a woman's woi-k, 
is 1{ ; or a woman's work is to a man's work :: I : 1'. 
Hence 27 mens work = 27xl| womon"s work 3 then, in 
place of Haying — 

21 women : 27 men : : 13 days : ? 

say tho work of 21 women : the work of 27xU r=33n 
3;J''xl3 ^- '^ 

%-=:20^« days. 



women : : 13.: 



_ 123. The ratio of the diameter of a circle to its 
circumference being that of 1 : .-J-Mlf)!), what is the 
circumference of a circle who.si! di;inu^ter is 47-3G feet ^ 
Ans. 148-78018 feet. 

124. If a pound (Troy wrl-lit) of .silver i,s worih (JGs., 





!H\ 






200 



RULE OF PROPORTION. 



whaj^is the value of a pound avoirdupoise ^ Ans. ^ 

ere^ltttaX"^^ ^'^^^^ to his 

16.. 35^. '''''^ ^'^ ^^^*- <^a" ie pay .? Ans. £m 

iei347 // t^^'feSl Z i"^"^"^^i^ «^ --1 costs 
Am. £1714 ihlUld ' *^' '°'* ^f ^« Irish mile.? 

127. If the rent of 46 aprpq q ,.««j i 
>s JeiOO, what will be the re^t'of '^^' ^"^ " P^oh^s, 
10 porches? Ans. ^vZTef, '' '"''"^' ^ ''^' ^ 

12 mL aday^B wl!: Sf'" .f/"'^^ "' '""e rate of 
him. How iy mUea a dir^'i^'p*^ '^y'' "^''^"k 
both to have started C„ ,hfl ? "■*^<''- *"»™g 

129. If the TOlue of Tn^ T° ^■^'f" ' ■^'"- 17. 
^£4 0,. 2j/ how manvTir avou-dupoise weight bo 
Pomd1roy'>AmclS' '^y ^o tad & one 

anf S;e\wf:batt:»t\" ' ^^'"'"« '" "« ---n'! 

'o' whaV : Se'w:!':?!:' \^'''™ "-- »» ^ 

jeiO 7s. 8J^. "'^ ""« ^^''ols garden? ^^j. 

;> ef da^s raXrii-'dt" i:t ? '* ■^^^f ' « 

three do it ? A71S. 2~i2 ^ ^^^* *^^® ^o«ld all 

« ^a,s , 1 aa, , , ,-,,■ --^^^ ^^^^ _ ^ ^^^^ ^^^^^ ^^^^^ 

«i<^.- = lCa, = : lwi,o,o„ft.e;7k'r^fp!rntro&:; 
#4-3 1. 3 __i44 7 °^7^^<^ C would do in a day 

finished in ^ davk'Tn"!*!'';*. .">"»""=■■' '-'-^l •>« 
"We to do U by hbuself?" ^i^i. Jr^Y^'^ ""»"f « 1« 



' ^ Atis. de4 

1871 to his 
'^212577517 
? ^%*. ^£30 

' canal costs 
Irish mUe ? 

14 perches, 
J roods, and 

the rate of 
's, overtook 
5l:i allowing 
47W. 17. 

weight be 
id for one 

lis tenant ; 

Dies to £4 
1 ^ Am. 

days; B 
would all 

le whole — 
3 in a day. 
le whole — 

> in a day. 
e whole — 

> m a day. 
in a day. 

ivrork :: 1 
le work) : 
Fit. 

B in 6^ 
'• will be 
d C be 



RULE or PROPORTION. 



201 



• ■ A, B, and C's work in one day=£ of the whole=|j|J 

Subtract- j A's work in 1 day=JV I _i i o of tha whole- **» 
ing j B's work in 1 day=/j. j -^s^ ^^ *^® wnoie-y^^, 

C'8 work in one day remains equal to . . . -^^^^ 
Then -f^-}^ (C's work in one day) : 1 whole of the work : : 1 
day : 2 ^i|, the time required. 

134. A ton of (Jbals yield about 9000 cubic feet of 
c;as ; a street lamp consumes about 5, and an argand 
Murner (one in which the air passes through the centre 
of the flame) 4 cubic feet in an hour. How many tona 
of coal would be required to keep 17493 street lamps, 
and 192724 argand burners in shops, &c., lighted for 
1000 hours? Ans. 95373^. 

135. The gas consumed in London requires about 
50,000 tons of coal per annum. For how long a time 
would the gas this quantity may be supposed to pro- 
duce (at the rate of 9000 cubic feet per ton), keep one 
argand light (consuming 4 cubic feet per hour) con- 
stantly burning } Ans. 12842 years and 170 days. 

? -* 136. It requires about 14,000 millions of silk worms 
to produce the silk consumed in the United Kingdom 
annually. Supposing that every pound requires 3500 
worms, and that one-fifth is wasted in throwing, how 
many pounds of manufactured sill, may these worms 
be supposed to produce ^ Ans. 1488 tons, 1 cwt., 3 qrs., 
17 1b. 

137. If one fibre of silk will sustain 50 grains, how 
many would be required to support 97 tb } Ans 13580. 

• 138. One fibre of silk a mile long weighs but 12 
grains ; how many miles would 4 millions of pounds, 
annually consumed in England, reach } 

Ans. 23333333331 miles. 

139. A leaden shot of A\ inches in diameter weighs 
17 lb ; but the size of a shot 4 inches in diameter, is to 
that of one A\ inches in diameter, as 64000 : 91125 : 
what is the weight of a leaden ball 4 inches in diameter > 
Ans. 11-9396. 

140. The sloth does not advance more than 100 









How loRij would it 



f o irn 



to 



IW! 



1 f r 



im 



Dublin to Cork, allowing the distance to be 160 English 
mil©8 ? Ans. 2816 days; or 8 years, nearly. 



if 



'i 



i „ I 



il ^:i\ 



202 



COMPOUND PROPORT/ON. 








141. li'ugliish race horsea l.^vn i. i 
tl^oruto of 58 miles ai W t "^ '^ ^^ ^' "^ 

vclocify, „ii..i,^ *i,/ 1" 'loiii. In what time at n;. 

-i-ir'Xrt^ ^"^ll*'"'^ ^'* 3000 tons; 

'^^or^^lZti tii r-""" ^'rs'- ^''"ut the 

tlioii luto hair-smin J """i'^ny, mado into steel a,„l 
w-te, there aTo X3''Z"« /^'! '''''- "'"J-th" 
gnuus of steel? ^,„ Sjooa™ ''°° "'""" ™"3 

COMPOUND PEOPOKTfON. 
-^^rfp^^^^^^^^^^^^ «. although t.o" 

i»'r."ft.t ratio „; th; thi.tt.erofl''''''"SH? '« «.e 
. jr. I'ut down the term, nf ? 1 . P™P""'''»n- 

■"«.e first and second ace, if "V'" """^ ^••"™ 
antecedents may form one n!f ™'','' " ^V that the 
mother In ^ttin^ Zu 7,7' "I- "'" »™«eque„t: 
oflect It has upon thf ansZ-if „?""' ""^M" "hat 

'^' - ^-w man, JJ^ ^:^^^^'^^^;;J« ^ a wall in 20 



icily 



i'^^^own{.32J,wiJlbea;a 



Hows 



m 



COMPOUND PRoroirnoN. 



'J03 



^n to go at 
^0, at til id 
Cork bo 

^000 tons ; 
' of 5 loet 
of 2 feet 

about tliG 
about ten 
pound of 
steel, and 
Joductiiic/ 
t>ut 7000 



Jgli two 
stion, to 
?>?• In 
ios, one 

to the 
I. 

ratios 
at the 
quents 

what 
idown 

anto- 
rm as 

livide 

in 20 

I 

'3 • 



ir • k \ '^o'^^'tiwis which givo 2U days. 
20 days imperfect ratio. 

1 days, the number sought. 
17 men 



37 yards 



conditions which give the required number ol" Unys. 





17 : 5 : : 20 : ? 
10 : 37 


And 17 
10 


5..oo.20x5x?>7 
3y 17x10 



'i'ho imperfect ratio consists of days — thoroforo we ar«i to 
(lilt 20, the given number of days, in the tliird place. Two 
ratios remain to be sot down — that of numbers of 7ncn, and 
that of numbers of yards. Taking the former first, wo ask 
ourselves how it affects the answer, and find tliat the more 
men there are, the smaller the required numlierwill be— tsinee 
the greater the number of men, the shorter the time ro{]uirod 
to do the work. We, therefore, set down 17 as anlecedeut, 
and 5 as consequent. Next, considering the ratio consisting 
of yards, we find that the larger the number of yards, the 
longer the time," before they are built — tlieroi'ore increasing 
their number increases the quantity retiuired. Hence we 
put 37 as consequent, and 10 as antecedent: and the whole 
will be as follows : — 



=13-0 days, n(;ariy. 



45. The result obtained by the rule is the siinio .'is wonM lie 
found by taking, in succession, the two j)roportioa8 supposed 
by the question. Thus 

. 1 5 men would build 16 yards in 20 days, iu how )uany 
'8 woeld they build 37 yards ' 

.(3' : 87 : : 20 : ^" — number of days which 5 men would 
16 

require, to build 37 yards. 

00 v37 
If 5 men would build 87 yards in.tl_r2 — days, iu how many 

16 
days would 17 men build them ? 

17 : 5 : : ?^ : 2^x5-17=20x5x37 ^^^ ^^^^^^ 
16 16 17X10 

of days found by the rule. 

40. ExAMPLK 2. — Tf 3 men in 4 days of 12 working hours 
each build 37 perches, in liow many days of 6 working 
hours ought 22 men to build 970 perches '.* 



204 




22 

8 
87 



8 :: 

12 
970 



COMPOUND PROPORTlOJf. 

3 men. 

4 (iixyH. 
■12 hours. 
37 porches. 

? days. 
8 Jiours. 
22 mon. 
970 porches. 



" • " i'ui uiies. 
3X12X97 0x4 
22X8 X a/ """=21 i days, nearly. 



days^U^islhtlmport't ?.r """''^fore 4 

place The more moZhXl^^ S.o^5^ '' P"' '" "»« ^'"rd 
form the work : therefore 29 jT * « *^''^*' necessary to per- 
smaller the nu^Cof worW E"* ^''?*' T*^ ^^ «««^""^I- K 
the number of days • hTnce 8^is Z fi *?' ^7' *^« ^'^^g'' 
The greater the number of perche?th^'*' ¥ ^^ «««""J- 
of days required to build tWn ^^'^ ^''^^^^ter the number 
put first, and 970 second ' consequently 17 is to bo 

or one in the first, a^ne in tL'"?! '^l? '^^"^ P^^«« J 
same number. ' '"^ *^^ ^^^''^ place, by tho 

Example 1 Tf ♦! 

32 : IGO • • 8 • ^^^^X20x8 ^ 



5 : 20 



32x5 



Dividing 32 and IfiO ^« qo , 



1 
1 



5 
4 



8 : 5x4x8=100 



measure a quantity in the S Z ''°^*?''' -"""^^^^ ^"' 
place ; or one in the first on?' ^"l^°o*her in the second 
This will in some iistanif 1 ^"°*^'' ^^ *^^^ ^^^'^ place 

into unity-^wSrrsrtr^^^^^^^^ ^-'^ti:- 



COMPOUM) PROPOKriON. 



205 



irly. 



therefore 4 
n the third 
^'^'•y to ncr- 
cond. The 
the larger 
12 second, 
ho number 
7 is to bo 



hy divid- 
id place ; 
J by tho 

iles Costa 
es cost ? 



uotients. 
pr{)|)or- 



!is long 
)er will 
second 
' place 
a^ntities 



ExAMPLK 2— If 28 loads of Htono of IS^wt. each, build a 
wall 20 foot lon^ and 7 foot hip;h, how nian\ loads of lU cwt. 
wjll build one 323 feet long and 9 feet high ? 

: 28 : 15x323x9x28 _^^^_ 



19 
20 

7 



15 : 

323 

9 



19X20X7 



Dividing 7 and 28 by 7, we obtain 1 and 4.— Substitutine 
those, we have ° 

19 : 15 : : 4 : 1 

20 : 323 
1:9 

Dividing 20 and 15 by 5, tho quotients ore 4 aai 3 : 

19 : 3 : : 4 : I 

4 : 323 

1 : 9 
Dividing 4 and 4 by 4, the quotients are 1 and I : 
19 : 3 : : 1 : ? 

1 : 323 

1 : 9 

Dividing 19 and 323 by 19, tho quotients are 1 ,ind 17 : 

1 : 3 :: 1 : 3x17x9=459. 
, . 1 : 17 

1:9 

In this process we moroly divide the first and second, or 
first and third terms, by the same number — which [29] does 
not alter the proportion. Or we divide the numerator and 
denominator of the fraction, found as the/oMr<A term, by the 
Kame number— which [Sec. IV. 15] does not alter the quo- 
tient. 



EXERCISES IN COMPOUNB PROPORTION. 

1. If £240 in 16 months gains £64, how much will 
d£60 gain m 6 months ? Ans. £6. 

2. With how many pounds sterling could I gain 
£5 per annum, if with £450 I gain £30 in 16 months ? 
Ans. £100. 

3. A merchant agrees with a carrier to bring 15 cwt 
of goods 40 miles for 10 crowns. How much ought hi 
to pay, in proportion, to have 6 owt. carried 32 miles ) 
Ans. IGs. 

K 2 



ii 



at 



U 



ii i^. 



:lf^«lf 



P 






■^f)(j 



<o.ui.oi;nd phopoijtion. 



Am. £20. ^'^'^^J " ^'"'"'''^ lUO iiiilos » 

fo/q/^ f^^^ "' ''^ ^^^«»-cIiandl.so aro carriod 40 ,niln« 
for Js*^ itV/"^.^^'^""^^ "''^^''^ ^« -rried 60 S« 
fn. T 1^^^ * ^^ inorchandiso aro carried 9n ,«{i 

1« honest™ Sr ie&AloO »r h' "'^* '" « ^'^^ 
would bo rcouimd f,^ 1 ''^ " ; """^ """V lioraM 

days ? AnXof ^'"'^ """^ *'"' """^'"'^ '« 3 

bein« i J toThtCS^^^^^^^^^^ wag.' 

men's, and 24 paiv of won.onVIlmt' 1 P"'"' °^ 

each kind woiJd ic T 1 '''"'*'> V"" many pair of 

13. A wall is r t,!' rl /■',"■ f women's shoes. 

how'^JI/iJrri/d'S i^ZT "''"T' u 

tunoei ,8 jays.^ .^„, X',^- tons the sa»o d.- < 
10. 11 ^/j,-. are the wa.ws of 4 «,or, /•... - ^^ 



wages of 



i^sn iOf 






I 



COMPOUND PROPORTIOM. 



307 



what Jill bo tlio wages of 14 iiiea for 10 day^P Am. 
16. If 120 busliols of corn List U horses 50 davs 

•7 

• iu!^ ^^ '' ^?°^!''^" ^'"''^^^ ^^^ "»'^« i" 3 days when 
the days arc 14 hours Jong, in how many day«ot'7 hours 
each will ho travel 300 nriles ? Ans j^ " '^*' ^' ^ *'°"'« 
• i^'o^/ the price of 10 oz. of bread, when the corn 
,s 4.. 2d per bushel, be 5^., what wm.st'be paid ibr 3 b 
12 oz when the eorn is 5s. r,d. per bushel ? ^1... 3 . 3/. 
VJ. 5 compositors m 16 days of 14 hours lon^^ can 
compose 20 sheets of 24 pages in each she , 50^11 .t" 

days ot 7 hours long may 10 compositors compose a 

Bltr filV'"-'^^ ^"/^^^ """^ ^^^*«^' containi^ng 40 
Bhects 16 pages in a sheet, 60 lines in a pa-e, and 
50 letters m a line ? Avs. 32 days ° ' 

M^^p}^ ^'^' been calculated that a square degree (about 
69X69 square miles) of water gives off by cvinor - 
tion 33 millions of tons of water |.er day. Ylow mu 

laLVt 7'\ ^^Vr^P^'^'"^ ''' ''^'ole surface 
to be 14 square feet ; and that the barometer stands at 
31 inches ? Ans, 13 tons 19 cwt. 

QUESTIONS IN RATIOS AND PUOPORTIOX. 

1. What is the rule of proportion; and is it ever 
called by any otlier name ? [IJ. 

2 What is the difference between simple and com- 
pound proportion ? [30 and 421 

3. What is a ratio ? [7]. 

4. What are the antecedent and consequent ? [71 
o. VV hat IS an inverse ratio ? [8] . 

6 What is tlie difffironno betwn"r» "- -..:ii-_--.« 1 
and a geometrical ratio ? [9j. ^^•^luat 



1 r-1 




208 



COMPOUND PROPORTION. 



7. ftow can we know whether or not an arithmohV^l 
or geometrical ratio, is altered in value p [10 and^' f'^ 

other? [Lj' "'"' '^''''''^'^ '^^'''''^ ^^ *^"^« «f ^"^ 
9 What is a proportion, or analogy ? [Ul 
10. What are means, and extremes ? flSl 

i.l%I^L:nir^^P'' - Soometial .oan of 
Jfc^roporL'rffe]''^' ^°« •!-»««- - - arith- 
miLKri^T™lf=" f- ■!-««. a.e in gco- 
fo^M'^llyZii^'" P"P°'"""^' '^ three quantUie, 

madt i!?T'"° the principal changes which may be 
made^m a geometrical proportion, without destroying 

r^li^;^"!? ^^ ?ecessarj, or even correct, to divide the 
rule of three into the direct, and inverse .P [35] 

18 How IS the question solved, when the first o, 
second terms are not of the .same denomination • or one 

19 hI-^''^ ''^*^^'^ ?^^^^^"* denomination ? [371^ 
if I' ll^Z '' ^ ^""'^^''^ ^" *^^« rule of proportion solved 
rf the thn^ term consists of more tha'n KeVoS 

.J^f' ^^"""^ i^i*' solved, if fractions or mixed numbe^-s 
oTin^rth^tet3%ra» ^^ ^"^-^^^ 

„f ^!' ^''° ^"^ °^ ""^ '''™« °f a question in the r„I„ 



1 1 'ii 



f 



209 



A H I T JI M E T I C . 



PART II. 



SECTION yi. 

. PRACTICE. 

.t'»'SS jJis" '""'"•"s «.■ —.J 

Mie latter '' ^ '^ *^'' ^'""'^" ^« *« "le price of 

tlip ni-Loa '41 parts, and bndiaj' the sura of 

« nart, " ,;"/n- P"',"' •"■ ''y dividiug^tho price iZ 

,>ot mJZTlZf ^^ a number, are those wf.ioh do 

any TateJ^ /o'^^t^'^idt: V-To'l T"''"^'^ "^ 
Mwe have seen fSeo It 2fiV,L f^?"' P"'''' ""' 

3 To find ti,t „r T J' „ "■* "'"t^'' measure it. 

B „, . ^- • ? • T"' P"'^ of "ly number- 
ing Quofaflf.; \I ^l"?'' divisor, and the result. 

fmtjU ; :,;! rtd^^ct „re;!::r:\':e ;7r 

gte'n nurab:;:' "° ''^ ""'""""^ all «„t prrS'e " „' 



810 



PRACTICE. 



parts^f 'ST""""^^^'"^ "'' *^'° ^^'""^' ^^^ «o«^Pound aliquot 

2)84 
2>!2 ' 
3)21 
7)7 

The prime aliquot parts aro 2, 3, and 7 ; and 
2x2= 4^ 
2x3= 6 
2x7=14 

2x2x3=12 [ "^''^ "'° compound aliquot parts. 

2x2x7=28 

2x3x7=4'^ 

14:^21 *2ran'd42^'''^'' ^^""''^ '° '''^''' ^'' 2' ^' ^' ^' ^' ^2, 
^ 5. Wo may apply this rule to appUcate numbers— Let it 

2)240 
2)jl20 

2)G0 

2)30 

3)15 
5)J5 

2X2= 
• 2x3= 
2x5= 
2x2x2= 
2x2x3= 12= 
2x2x5= 20= 1 
2x3x5= 30= 2 
2x2x2x2= 16= 1 
2x2x2x3= 24= 2 
2x2x2x5= 40= 3 
2x2x3x5= 00= 5 
2x2x2x2x3= 48= 4 
2x2x2x2x5= 80= fi 
2x2x2x3x5=120=10 



4 
G 

10 

8 



d. 



8 

6 

4 



4 





8 





in shillings 



PRACTICE. 



211 



And placed in order- 
£> d. 

3 



1 

1!0 



¥(5 

I 

4^? 



' = 4 



' = 5 



tV= 16= 1 



it 



d. 

4 
1 8 

24;= 2 



V= 8 



2^=12=1 



d. 




1= 30= 2 6 

1= 40= 3 4 

i= 48= 4 
i= GO 



5 



A= 80= G 





8 



I i=120=10 

Aliquot parts of a shilling, obtained in the same way- 



1 1 

T? — T 

_l I 

2? 2 

_l J 



s. d. 



Aliquot parts of avoirdupoise weight— 



s. d* 
4=6 



Aliquot parts of a ton. 

ton cwt. or 

I ; 1 9 

?o — 2 — ■^ 

20 ■■• ^ 

TIT— ^X= «5 

tV= 2 = 8 
J= 2.!=10 
^= 4"=16 
^= 5 =20 
J=10 =40 



Aliquot parts of a cwt. 
cwt. K) 



'=2 



! 

2? 



■» = 7 



8 



T? 

i=14 
4=16 
i=28 
1=56 



Aliquot parts of a quarter 
qr. ib 



A=2 
1=14 



, Aliquot parts may, in the same manner, be easily 
obtamed by the pupil from the other tables of weights 
and measures, page 3, &c. ^ 

6. To find the price of a quantity of one dcnomina- 
tion— the price of a " higher" being given 

Rule.— Divide the price by thai number which ex- 
presses how many times we must take the lower to 
make the amount equal to one of the higher d'enomina- 

pei^^cwrr'~^^''* '' *^^ P''"' ^^ ^"^ ^^ ^^ ^"**^^ ^t '2.. 

Tl^^f'''"!*^^^^^ ^\^^ ^ «*""^ « ti"^es, to make 1 cwt. 
Therefore the price of 1 cwt. divided by 8, or 72s -^S-oI 
IS the price of 14 ib. ^ , ux <^. . o—y*., 

The table of tiliquot pn'^^ of avoirdimms,> ^xo^r-ht -hr^s, 
ti;f pile'' o'F l*ti"'' " ^' "• ^■'■'•-f"-"'" F- i*^ '!>; J rf^ 






lis 



ill 



212 



PRACTICE. 




1. 

2. 
3. 
4. 



EXERCISES. 

What is the price of 
i cwt. at 29s. 6d. per cwt. ? Ans. Is. 4^d 
,- a yard of cloth, at Ss. Gd. per yard ? An!. 4s. 3d 
14 ft, of sugar at 45.. 6d. p.r cwt. ? Ans. 5s. S^d 
What IS the price of ^ cwt., at 50.. per cwt. > 

£ s. d. 
505.=2 10 



qra. cwt. 
The price of 2=i is 



of 1=1-^.2 



IS 



] 





£> s. 
0=2 10-^2 
12 G=l 54-2 



5 



. Therefore the price of 2-f 1 qrs.(=| cwt.) is 1 i7g 

J« half the price of 2 qrl'' Th&.rth " pi^'e'of I'llTu 
onf ow" ^"'' '' ' '^'- P^"^ *^- ^^^ «f '^"f tlfe ^ile of 



6. 

6. 

7. 
8. 



. What is the price of 
? oz- of cloves, at 9.. 4rf. per lb ? Am. 3id 
nail of lace, at 15.. 4d. per yard ? Ans."" ' 
2 10, at 23s. 4d. per cwt. .? yl?^.?. ii^ 
^ ib, at 18.. Sd. per cwt. ? Ans li^." 



14^/. 



7. When the pric3 of wwrc ^A^w o^j^ 'qow->r'' dono 
minatiou is required— ^^^ 

Ia«f vnf;""^'^^^^' P""' °^ '^^^^^ denomination by the 
last rule ; and the sum of the prices obtained will be 
the requu-ed quantity. [ '^^ 

at «;.;•:? c-t'"?^' " "" P™= °f 2 <!-■ 1* » of -gar, 

s. d. 
.45 price of 1 cwt. 

cwt A-nAoo A ti- ^ . [or ,1 of 1 cwt. 

14\b=f,or'-of2or8 ^■^' ^^ ^•^••-^/=22.. G,i.4-4, is the 

s, or , 01 ^ qrs. p^ce of 14 lb, the i of 1 cwt.. 

\ ^^i^~T; . 01^ the { of 2 qrs. 
2nr« t n . i H IS the price of 2 qrs. 14 lb. 
^ ^J^«-rT5 ?f 1 cwt. Therefore 45.^ (f,li« r^vj^. of 1 o.-f ^ - o 
or zos. Qu., 18 the price of 2 qrs, ^' ' o^,t.J~2, 



PRACTICE. 



213 



Ans. 4s. 3d 
Ins. OS. S^^d 
r Gwt. ? 

d. 


£ s. 
0=2 10-^2 
_G=1 5h-2 

G 

\vt. : md ita 
nd its price 
of f cwt. is 
lie i^rice of 



. S^d. 
ns. llid. 



3r" dcno- 

on by tlie 
d will be 



of sugar, 



of 1 cwt. 
Bof2qrs., 
^4, is the 
of 1 cwt., 

14 lb. 

nTt.)-f2, 



45^^ JoJ f, ^ rV ^i^.y^^i^f 2 qrs. Therefore 
T rJo ? ; f?'- S'f7-'^=5*- 7Arf., is the price of 14 !b. 

f; mT. • l"^^'--^^'^-'.^.' *^^ P'^^*'^ ^f 2 qrs. plus the price 
ot 14 lb, IS the price of 2 qrs. 14 lb. i i- f 

EXERCISES. 

What is the price of 

9. 1 qr., 14 fe at 46^. 6d. per cwt. ? Ans. 17s. 5id. 

10. 3 qrs. 2 nails, at 17*. 6c?. per yard ^ Ans 
los. 3|r/. 

11.5 roods 14 perches at 3s. lOd. per acre ? Ans. 
5s. l^d. 

12. 16 dwt. 14 grs., at £4 4s. 9d. per oz. ? Ans 
£3 10s. 3}d. ^ 

13. 14 lb 5 oz., at 25.5. 4d. per cwt. ? Ans. 3s. 2fd 

8. When the price of ow, "higher" denomination is 
required — 

KuLE. — Find whaf ninnbor of times the lower deno- 
mination must be taken, to make a quantity equal to 
one of the given denomination ; and multiply the price 
by that number. (This is the reverse of the rule eiven 
above [G]). ^ 

ExAMPLK.— What is the price of 2 tons of sucar, at 50s. 
per cwt. < 

1 \?.^'^'^' ^^ *^® ^'« ^^ 2 tone : hence tlie price of 2 tons will 
be 40 times th price of 1 cwt.— or 50,9.x40=£100. 
50.S-. the price of 1 cwt. multiplied 
^y 40 the number of hundreds in 2 tons, 
gives 2000,s-. 

or XlOO as the price of 40 cwt., or 2 tons. 

EXERCISES. 

What is the price of 

14. 47 cwt., at l.y. S^. per lb I Ans. £438 VSs. 4d 

15. 36 yards, at 4d. per nail r Ans. £[) V2s. 

16. 14 acres, at 5s. per porch .? Ans. £5f)0. 

17. 12 R), at l|f/. per grain ? Ans. £504. 

IS. 19 hhds., at 3d. per gallon : Ans. i214 19*. 3^. 

0. When the price of more Ihitu one "higher" dcno- 
miuatiou is required — 



ill 



214 



PRACTICE 



• RuLE.—Find the price of each bv the lasf nn^ nrU 

atlTr"o"^eT^* " *'^ ^'^^ <^^ ^ «-*• ^ <1- of flour, 
1 stone is the j\ of 2 cwt. Therefore 

tv,„u- r ^ 1. ,?^'-'*^o price of one stone, ' 
multiphed by B, the number of stones in 2 cwt, 

gives 3a?., ;• ^?ce ot 16 stones, or 2 cwt. 

EXERCISES. 

in e , What is the price of 
.£1 L ^ ' ^ ^''•' ^ °^"'' ^* ¥' P«^ ^^« -^ ^^^. 
20. 6 cwt. 14 ib at 3^. per ib ? Ans. £8 Us. 6d. 
^1. 3 ib 5 oz. at 2id. per oz. ? Ans. 9s. lUd. 

ul^fieMTlfc.' ^^^^«>3P-^^-> at 5. per perch .P 

fini^/i. ^^^"^ *^.^ P"^^ °^ ^"^ denomination is given to 
find the price of any number of another— ^ ' 

±tuLE.-.Find the price of one of that other denomi- 
nation, and multiply it by the given number of the 

^^XAMPLE.-What is the price of 13 stones at 255. per 

1 stone=| cwt. Therefore 

8)25^, t he price of 1 cwt. divided by 8, 

we obtain £2 7^ M the price of 13 stones. 
Istoneistliejoflowt. Hence 25.!.^8=3s 1 u i, th. 
pnoe of one stone; and 3,. lirf.xlS, the price of is '.it! 



PRACTICE. 816 

EXERCISES. 

What is the price of 

24. 19 lb, at 2d. per oz. .? Atis. £,2 \0s. Sd. 

25. 13 oz., at \s. Ad. per ft) ? Am. \s. \d. 

26. 14 ft), at 2s. Qd. per dwt. ? Ans. &420. 

27. 15 acres, at 185. per perch ? Ans. ^£2160. 

28. 8 yards, at Ad. per nail ? Ans. £2 2s. Sd. 

29. 12 hhds., at 5d. per pint ? Ans. ^£126. 

30. 3 quarts, at 91^. per hhd. ? Ans. Is. Id. 

11. When the price of a given denomination is the 
aliquot part of a shilling, to find the price of any num- 
ber of that denomination — 

Rule. — Divide the amount of the given denomina- 
tion by the number expressing what aliquot part the 
given price is of a shilling, and the quotient will be the 
required price in shillings, &c. 

Example. — What is the price of 831 articles at 4d. per } 
3)831 

277s.=£13 17s., is the required price. 

Ad. is the | of a shilling. Hence the price at Ad. is i of 
what it would be at Is. per article. But the price at Is. per 
article would be 831s.:— therefore the price at Ad, is 831s. -^ 3 
or 277s. 






lii 



EXERCISES. 

What is the price of 

31. 379 ft) of sugar, at 6d. per lb ? Aiis. £9 9s. 6d. 

32. 5014 yards of calico, at 3d. per yard.? Ans. 
£62 13s. 6d. 

33. 258 yards of tape, at 2d. per yard ? Ans. £2 3s. 

12. When the price of a given denomination is the 
aliquot part of a pound, to find the price of any number 
of that denomination — 

Rule. — Divide the quantity whose price Is sought 
by that number which expresses what aliquot part the 
given price is of a pound. The quotient will be the re- 
quired price in pounds, &c. 



216 



PRACTICE. 



^ J«Mr,..._What is tI.o price of 1732 ft of to., .t 6, 
would bo jeiT-^p fi. r P°.^ ^^- ^^^^ at ^1 per lb it 



EXERCISES. 

Wliat is tho price of 
35. 13 oz., at 4.. per oz. ? Ans. £2 12s. 
37. 83 a, at Is. Ad. per ib ? Ans. £5 10s Sd 
39* 976 r-V'f o'^- P°^ ^^- -^ ^-- ^3 16.. si. 
4?' "? »»' ^* ^^>^- pel- ft ? ^.^..^2 6.. 8^ 

44. 1000 ib, at 35. Ad. per lb .? ^w.. jgieg 13,. 4J. 

13. The complement of the Drice i«i wliof ;f ,„„ * i. 
pound or a shilling ^ ^* ^* ^^^*^ ^^ » 

por^."d r-"^''"* '•'"'» Prf^o "f 1«0 yard., at 13», «. 
Cs. Sd. (the complement of 13s. 4^-) ia 1 of £1 

Z" f 'S *■>» Pri"* »t *1 per yard 
.uhtraot ^, the ^rioo at fe. sS. (tCoomplement) 

andthedi£fe™oo, 980, will ho the price at ^^Spor yard. 

I47^™tTlf\t]^:.1;:;' t'J^O at C. 8. are e,„al to 

price of 1470 at^lSs, «34hepr1oe^ofl?70 at £?'"'•*''' 
.the price of 1470 at Gs. &(. per yard ' '°"" 



PRACTICE. 



217 



..■i 



tea, at 5j 

2 lb is tht, 
1 per lb it 



> 135. 4d. 

12 7s. 6d. 

Sd. 

8d. 



3 9*. 2d. 

• 

13*. 4d. 

tnts of a 

uot part 
lot — 
ho caso 
IcuJated 

13s. 4cl 
1. 

lement) 

3r yard. 

qual to 
ice the 
I minus 



EXERCtSEiS. 

What is tho price of 

45. 51 ib, at 175. 6^. per lb .? Ans. £44 125. 6a 

46. 39 oz., at 7d. per oz. ? Ans. £1 2s. 9d. 

47. 91 ft), at lOd. per ib ? Ans. £3 15s. lOd. 

48. 432 cwt., at 165. per cwt. } Ans. £345 12s. 

14. When neither the price nor its complement is 
the aliquot part or parts of a pound or shilling — 

Rule 1. — Divide the price into pounds (if there are 
any), and aliquot parts of a pound or shilling; then 
find the price at each of these (by preceding rules) : — 
the sum of the prices will be what is required. 

Example.— What is tho price of 822 lb, at £5 19s. SJtZ. 
per lb I £5 m. ^d.=£6+m. Zld. 

s. d. £> 

8 —I 



!fM 



But 195. ZM. 



< 



10 
G 
2 





0=1 



or jV of the last 
or ^ of the last 



Henco tho price at £5 195. Z'^d. is equal to 



£ 

822x5 

82a 



£ s. d. 
=4110 0, tho price at 



822 

3 
8 22 

It 



)l22?^123_i_ A\ 



= 411 
= 274 
= 102 15 
= 5 2 9 
= 17 14 



£ s. d. 

5 per lb. 



£i or 10 
£l or 6 8 



£] or 2 6 
;; £ I ov 1-J 

;;4i;oro ooi 



11 
11 



And X4903 14 lOi is the price at £5 19 3 J „ 

The price at tho whole, is evidently equal to the sura of 
the prices at each of tho parts. 

If the price were £5 19s. 3^d. per lb, we should sub- 
tract, and not add the price at }d. per lb ; and wc then 
would have £4902 05. l^d. as the answer. 

15. Rule 2.— Find tho price at a pound, a shilling, 
a penny and a farthing ; then multiply each by thoir 



[ , 



II 



f '! 



t 



iJI 



n 



m^:^' 



218 



PRACTICE. 



produl'' Tf""^' m' '" '^' Sivon piico ; and add the 
pioaucta. Using the same example 

£ ». d. £ a 

^llui S S5j;»®P^!«eat£l)x 5=4110 6 

Its Q X (*?0P^'ceatl«.)xi9= 780 18 

^ 1? J.Sf'oP'-Jceatle/OX 3= 10 5 

i7 li(tho price at 4flf.)x 8= 2 11 



d. 

the price at £5 
» 19*. 
6 » 8d 

4i „ id. 



And the price at £5 Ids. 3|rf. is ^£4903 14 lOi 

tlie^hi.w''/'"-^^^. *^' P^'^°" ^' *»^« "«^t number of 
the highest denomination ; and deduct the price at tlia 
difference between the assumed and given prFce 
Usmg stiU the same example— 

prfco.'' '''''' *' ^^-'^^ ^'^^''^ denomination in the givea 

From the price at £6 ^ "' ^' nr 4?qo n n 
Deduct the pri^e( the price at 8rf.=27' 8 > ^ ^ ^ 

"'*'''' < .. 4rf.= IZlijO' 28 5 U 



at 8id. 



The diflFerence will bo the price at £6 19*. 8| or i^lTm 
17 Rule 4.— Find the price at the next hijrher 

atT <^ff ' '^ ' Tf^^ '' ^""^"g ' ^-d deduct the p1:Se 
at the difference between the assumed, and given price 

ExAMPLE.-What is the price of 84 lb, at Qs. per tb^ 
ds.-^Gs. 8d. minus 8J.=^ minus i-^10. 

we have ^25 4 0, the price at 6s. 

EXERCISES. 

^r. ^o « ^^^^'^^ ^^ *^^ P^ice of 

49. 73 ib, at 13^. per ft ? Ans. £^7 9s. 

i>0. 97 cwt., at 15^. Qd. per cwt. ? Ans. £1Q 7s. 



51. 43 ft, at 3s. 2d. per ft ? Ans. £6 'l6s. 2d 



9d. 



r)2. 13 
OS. lid. 

53. 27 yards, at 7*. ' 5U. 
Is. llid. 



acres, at £4 5s. lid. per acre.? Am. £55 



per yard.? Ans. JBIO 



llsTtl t' p™' '^ "" "™ °""»''^' °f ^'i»S». 



PRACTICE. 



219 



add tlio 



>rioe at £5 
„ Ids. 
8d 
id. 



imber of 
30 at tlid 



lie given 



a. 
i2 



d 




!8 5 li 

3 14 lOi 
higher 
he price 
n price 



pep lb. 



7.1. 9d. 
d. 

. £10 
illings, 



Rule. — Multiply the number of articles by half the 
number of Hhillings ; and consider the tens of the pro- 
duct as pounds, and the units doubled^ as Hhillings. 

Example. — What i{ the price of G4G lb, at 16s. per lb 1 

646 
8 



510 



£510 10,s. 

2s. being the tenth of a pound, there are, in the price, 
half as many tenths as shillings. Therefore half the number 
of shillings, multiplied by the number of articles, will express 
the number of tenths of a pound in the price of the entire. 
The tens of these tenths will be the number of pounds ; and 
the units (bcigg tenths of a pound) will be half the required 
number of shillings— or, multiplied by 2— the required num- 
ber of shillings. 

In the example, 10.9., or £-8, is the price of each article. 
Therefore, since there are 040 articles, 040xi^8=ii5.10-8 
is the price of them. But 8 tenths of a pound (the unitHn 
the product obtained) , are twice as many shillings ; and henco 
we are to multiply the units in the product by 2. 

EXERCISES. 

What is the price of 

54. 3215 ells, at 6s. per ell.? Ans. £964 lO.v. 

55. 7563 lb, at 8s. per lb } Ans. £3025 4.-. 

56. 269 cwt., at 16s. per cwt. } A"'!. £215' 4s. 

57. 27 oz., at 4s. per oz. .? Ans. £ 8s. 

58. 84 gallons, at 14s. per gallon ." Ans. £58 I6s. 

19. When the price is an odd number of shillings, 
and less than 20 — 

Ruj^E. — Find the amount at the next lower even 
number of shillings ; and add the price at one shilling. 

Example.— What is the price of 275 lb, at 17s. per lb ? 

8 



220 
13 15 



The price at 16s. (by the last rule) is 
The price at Is. is 275s.= 

Hence the price at 16s.+ls , or 17s., is £233 15s. 




i 



Ik 



Ih 



220 



PnACTICB 



Tho prico at 17.v. is equal to tlio 



prioe at oiio aJjiliing. 



price at IGs., plus the 



EXERCISES. 



59. 86 oz., at 5*. per oz. ? Ans. £'21 10.? 

60. 62 cwt at 195. por cwt. ? Ans. £i)S ' ISs. 
ro ^^yf^^'^^ 175. per yard.? yl^i^. X'll is*. 
02. 439 tons, at 11^. per ton ? Ans. jC;241 9.v. 
Od. 96 gallons, at 7s. per gallon ? Ans. i233 12^. 

number-'''' *^° ^""""^'^^ '' ^^P^-^^^^^^^'l l>y a mixed 
liuLE.— Find the prico of tho integral part. Then 

t7o"n t^d d- ^'ny^'' ^ «- nume?ato/of the fr J. 
ti^on and divide the product by its dcnoniinator-tho 
quotien will be the price of tho fractional part. To 
sum of these prices will be the price of the whole quan- 

^ ExAMPLE.-What is the price of ^ lb of tea, at 5.. per 



The price of8 lb is 8x5?.= 
The prico of |- lb is ^^'• 



£ f. (I 
2 







3 



9 



And the price ofS^ lb is . . 2~T~U 



of a 



EXERCISES. 



64. 
65. 
66. 
67. 
4id. 
68. 

4s. ed. 

69. 
iP2751 



What is the price of 

oIqIT""' ?* ^^- ^f' P*^^ ^°2«" -^ ^^^•'. 175. 101./. 
xi/dy It), at 25. brf. per lb .? ^^5. ^£34 3^ ] j^" 

5302 lb, at 145. pei: lb ? Ans. 371 IO5. 'e/ ' 

178f cwt., at 175. per cwt. ? Ans. i2151 125 

7023 cwt., at £1 12s. 6d. per cwt. .? Ans. £1239 

^\^^\.,?*-' ^^ ^^ ^'- '^^' per cwt..? Ar^. 
lis. b^d. 



PRACTICK. 



221 



s.f plus the 



lS,s. 

18*. 

9.V. 
!3 12s. 

a njixcd 

t. Then 
tho frac- 
%tOT — tho 
rt. Tho 
Die quan- 

at 5.'?. i)cr 



-)v\co of a 



s. 101(1. 
'. Ud. 
6(L 
51 125 

^1239 

' Ans. 



21. The rules for finding tho pnco of Bovoral deno- 
minations, that of one being given [7 and 9], may bo 
abbreviated by those which follow — 

Avoinliopoise Weight. — Given the price per cwt., to 
find the price of hundreds, quarters, &c. — 

lluLE. — Having brought tho tons, if any, to cwt., 
multiply 1 by tho number of hundreds, and consider tho 
product as pounds sterling ; 5 by the number of quar- 
ters, and consider tho product as shillings ; 2^, tho 
number of pounds, and consider the product as pence : — 
tho sum of all the products will be tho price at £1 per 
cwt. From this find tho price, at the given. number of 
poun '''=', shillings, &c. , 

Example.— What is the price of 472 cwt., 3 qrs., 10 lb, 

at £5 \)s. 6(1. per cwt. 1 

£ s. d. 

I 5 2'- 

Multipliers 472 3 16^ 

472 17 10| is the price at £1 per cwt. 
5 




2589 1 9| the price, at £5 9s 



At £1 per cwt., there will be £1 for every cwt. We mul- 
tiply the qrs. by 5, for shillings ; because, if one cwt. costs £1, 
the fourth of 1 cwt., or one quarter, will cost the lourth ot 
a pound, or 5s.— and there will be as many times 5s. as there 
are quarters. The pounds are multiplied by 2\ ; because it 
the quarter costs 5s., the 28th part of a quarter, or 1 lb, 
must cost the 28th part of 5s., or S^c/.— and there will be as 
many times ^d. as there are pounds. 

EXERCISES. 

What is the price of 

70. 499 cwt., 3 qrs., 25 fib, at 25s. lid. per cwt. ? 
Ans. £647 I7s. l\d. 

71. 106 cwt., 3 qrs., 14 ft), at 18*. M, per owt. ? 
Ans. jeiOO 35. 10|rf. 





PRACTICE, 

^I'-^mo 55'.'4.r- ' '^' =" '''■ '"■' P«' -'• ? 
jB4?i7l° eZ*'' ^ '■'''"■' '* '''' *' '*'• "''• P""' •='"■ ■' ^'"• 
jes^ss^sij''' ^ '^^^'' '' "^' "' ''™- "''• P" ™'- • ^'"• 

A«:£L r iwf/"' '" *' ^' '^'- '"■ p='- -'• •' 

^r-iri 7^/ '■■■' '' ''' "* '''■ ^- p°'- -'••' 

22. To find the price of cwfc., qrs., &c., the price of 
a pound being given — 

RuLE.—Haviug reduced the tons, if any, to cwt., 
multiply 9. Ad. by the number of pence contained in 
the price of one pound :— this will be the price of one 

"""^i?"!. I^^ *!'^ P"^"-' °^°"^ ''^^- ^'y 4, and the quotient 
Will be the price of one quarter, &c. 

Multiply the price of 1 cwt. by the number of cwt. • 
the price of a quarter by the number of quarters ; the 
price of a pound by the number of pounds ; and the sum 
o± the products will be the price of tlie given quantity. 

Example. 
8(/. per lb. ? 

d. 



-What is the price of 4 cwt., 3 qrs., 7 lb, at 



9 4 
8 



s. d. 



28 R « f ^ •'' ''H '"^^- Xj.^i" give 298 8 the price Of 4 cwt. 
28)18 8 e price of Iqr. X3,willgive 66 the price of 3 qrs. 
8 the price of 1 lb X7, will gi ve 4 8 the price of 7 lb. 

20)369 4 
And the price of the whole will be £17 l£~i 

At Id. per lb the price of 1 cwt. would be l\M. or 9i-. Ad. •— 
therefore the price por cwt. will be as many times 0*. U as 
there are pence in the price of .a nnnnd. The >iri-" -p - 
quarter IS \ the price ot 1 cwt. ; an<l there will bo as many 
times the price of a quarter, as there are quarters, &c. 



PRACTICE. 



223 



!l»'ifl 



EXERCISES. 

What is tlie price of 

79. 1 cwt., at 6d. per ft) > Am. £2 16s. 

80. 3 cwt., 2 qrs., 5 ib, at 4d. per Bb .? Atis. £6 
12s. 4d. 

81. 61 cwt., 3 qrs., 21 lb, at 9d. per Bb } Am. £2\B 
2s. M. 

82. 42 cwt., qrs., 5 ft), at 2bd. per lb r iln*. ^2490 
105. bd. 

83. 10 cwt., 3 qrs., 27 Bb, at bid. per ft) ? Am. 
£2Q\ Us. Qd. 

23. Given the price of a pound, to find that of a ton — 
EuLE. — ^Multiply £Q 6s. 8d: by the number of pence 
contained in the price of a pound. 

Example. — ^What is the price of a ton, at 7d, per ft) '? 

JS s. d. 
9 6 8 

7 



65 



6 8 is the price of 1 ton. 

If one pound cost Id., a ton will cost 2240rf., or £9 6s. Sd. 
Hence there will be as many times £9 6s. Sd. in the price 
of a ton, as there are pence in the price of a pound. 



EXERCISES. 

What is the price of 

84. 1 ton, at 3^. per ft) > Am. £28. 

85. 1 ton, at 9d. per ft) .? Am. £84. 

86. 1 ton, at 10c?. per ft) ? Am. £93 6^. 8d. 

87. 1 ton, at 4d. per ft) T Am. £37 65. Sd. 

The price of any number of tons will be found, if we mul- 
tiply the price of 1 ton by that ntmiber. 

24. Troy Weight. — Given the price of an ounce — ^to 
find that of ounces, pennyweights, &c. — 

Rule. — Having reduced the pounds, if any, to ounces, 
set down the ounces as pounds sterling ; the dwt. as 
shillmgs ; and the grs. as halfpence : — this will give the 
price at £1 per ounce. Take the same part, or parts, 
&c., of this, as the price per ounce is of a pound. 



224 



PRACTICE. 




Example 1 -What is the price of 538 oz., 18 dwt, 14 
grs., at lis. 6d. per oz. ? » , ^^ uwt , i^ 

Us. Gd. =il-L.^J4.ij^2 

£> s. d. 
2)538 18 7 is the price, at ^1 per ounce. 

^ol^fr 1 « i?f ^' *^® P^^^«' a* 10^- per ounce. ^' 
^ 1 Q n li }^ *^® P"^®' «-* !«• per ounce. 
•^^ '^ 5f 18 the price, at 6d. per ounce. 

And 309 17 8i is the price, at lb. 6d. per ounce. 
14 halfpence are set down as 7 pence. 

If one ounce or 20 dwt. cost £1, 1 dwt. or the 20th part 

-^4th part of 1 dwt., or 1 gr. will cost the 24th part of 
IS. — or ^o. ^ 

^E^xAMPLE 2.--What is the price of 8 oz. 20 grs., at .^£3 

£> s. d. 

8 10 is the price, at £1 per ounce. 
o 



Price It £1 ' 10-0 ifi ? • t5^ P"^®' ""^ 5^ P^^ ^^''e- 
vTnl I o A n . 1. i^ *^e P"«e, at 2s. per ounce. 
I nee at 2s.^ 4 =0 4 0^ is the price, at 6d. per ounce. 

And £25 2 7^ is the price, at ^£3 2s. 6(i. per oz. 

EXERCISES. 

What is the price of 
88. 147 oz., 14 dwt., 14 grs., at 75. 6d. per oz. .? 
-4715. £55 7*. 11^^. p "^.. 

, 89. 194 oz., 13 dwt., 16 grs., at 11*. 6d. per oz. ? 
Ans. ieill 18*. 10^(£. ^ 

90. 214 oz., 14 dwt., 16 grs., at 12*. 6d. per oz. } 
Ans. £134 45. 2d. ^ 

91. 11 ft), 10 oz., 10 dwt., 20 grs., at 105. per oz. ? 
Ans. £71 55. 5eZ. "^ 

92. 19 ft), 4oz.,3grs.,at£2 55. 2(^. peroz. .? ^W5. 
£523 185. lli<^. ■^ 

93. 3 oz., 5 dwt., 12 grs., at £1 65. 8<^. per oz. ? 
Ans. £4 7*. 3J</. 



llHi 



' ^i 



PRACTICK. 



225 



18 dwt, 14 



ice. 

ice. 

ce. 

je. 

' ounce. 

3 20th part 
. ; and tlio 
th paxt of 

5rs., at .-£3 



r ounce. 



per ounce, 
per ounce* 
per ounce. 

Qd. per oz. 



per oz. ? 
per oz. ? 
per oz. } 
per oz. ? 
? Ans. 
per oz. ? 



25. Cloth Measure. — Griven the price per yard — to 
find the price of yards, quarters, &c. — 

Rule. — Multiply £1 by the number of yards ; 55. by 
the number of quarters ; Is. 3d, by the number of nails ; 
and add those together for the price of the quantity at 
£1 per yard ? Take the same part, or parts, &c., of this, 
as the price is of iSl . 

Example 1.— What is the price of 97 yards, 3 qrs., 3 
nails, at 8s, per yard ? 

£1 5s. Is. Zd. 

MuItlpUers 97 3 2 

2 )97 17 6 is the price, at £1 per yard. 
5)48 18 9 is the price, at IO5. per yard. 
From this subtract 9 15 9 the price, at 2s. per yard. 

And the remainder 39 3 is the price, at 8s, (10s.— 2s.) 
If a yard costs £1, a quarter of a yard must cost 5s. ; and 
a nail, or the 4th of a yard, will cost the 4th part of 5s. or 
Is. 3fZ. 

Example 2.— What is the price of 17 yards, 3 qrs., 2 
nails, at £2 5s. 9d. per yard ? 

£1 5s. Is. Zd. 
Multipliers 17 3 2 

17 17 6 is the price, at £1 per yard 

35 15 is the price, at £2 per yard. 
The price at £l-^ 4=4 9 4i is the price, at 5s. 
Th© price at 5s. -^ 10=0 8 11| is the price, at 6d. 
The price at GcZ.-f- 2=0 4 5| is the price, at 3^. 

And £40 17 9^ is the price, at £2 5s. 9d. 

EXERCISES. 

What is the price of 

94. 176 yards, 2 qrs., 2 nails, a 15s. per yard > Ans. 
jei32 9s. 4^d. 

95. 37 yards, 3 qrs., at o^l 5s. per yard ? Ans, £47 
3s. 9d. 

96. 49 yards, 3 qrs., 2 nails, at £1 10s. per yard.? 
A.ns. £n 16s. 3d. 

97. 98 yards, 3 qrs., 1 nail, at £1 15s. per yard.? 
Ans. £172 18s. d\d. 



H 




226 



PRACTICE. 



98. 3 yards, 1 qr., at 17s. 6d. per yard.? Am £2 
16s. lOid. 

99. 4 yards, 2 q^" , 3 nails, at ^1 2s. Ad. per yard ? 
Am. £5 4s. S^d. 

26. Land Pleasure. — Kule. — Multiply £1 by tLe 
number of acres ; 55. by the number of roods ; and l^d. 
by the number of perches : — the sum of the products will 
be the price at £1 per acre. From this find tho price, 
at the given sum. 

Example. — What is the rent of 7 acres, 3 roods, 16 
perches, at J£3 8s. per acre '? 

£ s. d. 



Multipliers 



15 1^ 
7 3 16 



Sum of the products 7 17 0, or the price at JEl per acre. 



23 11 the price at £Z per acre. 
3 18 6 the price at 10s. per acre. 



27 9 6 the price at £3 10s. per aero. 
Subtract 15 8i the price at 2s. per acre. 



And 26 13 



Si 



,j is the price at £3 8s. 

If one acre costs £1, a quarter of an acre, or one rood, must 
cost 5s. ; and the 40th part of a quarter, or one perch, must 
cost the 40th part of 5s. — or Ikd. 



•'I n 



EXERCISES. 

What is the rent of 

100. 176 acres, 2 roods, 17 perches, at £5 6s. per 
acre ? Ans. iE936 Os. 3d. 

101. 256 acres, 3 roods, 16 perches, at £6 6s. 6d. 
per acre .? Ans. ig 1624 lis. 6id. 

102. 144 acres, 1 rood, 14 perches, at £5 6s. 8d. per 
acre ? Ans. £769 16s 

103. 344 acres, 3 rbods, 15 perches, at £4 Is. Id. 
per acre > Ans. £1398 Is. Id. 

27. Wine Pleasure. — To find the price of a hogs- 
head, when the price of a quart is given — 

Rule. — For each hogshead, reckon as many pounds, 
and shillings as there arc pence per quart. 



PRACTICE. 



227 



Am £2 
per yard ? 

1 by the 

and l^d. 

)ducts will 

tho price, 

roods, 16 



per aore. 



r acre. 
iv acre. 

t. per aero. 
V acre. 

.8s. 

rood, must 
erch, must 



15 6^. per 
'6 6s. 6d. 
s. 8d. per 
4 Is. Id. 

F a hogs- 
y pounds, 



Lkkvivve. — What is the price of a hogshead at dd. per 
quart? A)is. £9 ds. . 

One hogshead at Id. per quart would bo 63X4, since there 
are 4 quarts in one gallon, and G3 gallons in one hhd. But 
G;>x4d.=252<i.=<£l Is.; and, therefore, the price, at dd. per 
quart, will bo nine times as much — or 9X^1 ls.=£0 9«. 

EXERCISES. 

AVhat is the price of 

104. 1 hhd. at 18^. per quart ? Ans. £18 185. 

105. 1 hhd. at 19<-^. per quart? Ans. £19 195. 

106. 1 hhd. at 20d. per quart ? Ans. £21. 

107. 1 hhd. at 2s. per quart ? Ans. £25 45. 

108. 1 hhd. at 25. 6d. per quart? Aiis. £31 105. 

When the price of a pint is given, of coxirse we know that 
of a quarc. 

28. Uiven the price of a quart, to find that of a tun— 
Rule. — Take 4 times as many pounds, and 4 times 
as many shillings as there are pence per quart. 

Example. — What is the price of a tun at lid. per quart * 
£ s. 
11 11 
4 



4G 4 is the price of a tun. 

Since a tun contains 4 hogsheads, its price must he 4 tiinp,* 
the price of a hhd. : that is, 4 times as many pounds and shil- 
lings, as pence per quart [27]. 

EXERCISES. 

What is the price of 

109. 1 tun, at VJd. per quart? Ans. £79 IGs. 

110. 1 tun, at 2Qd. per quart ? Ans. £84. 

111. 1 tun, at 25. per quart ? Ans. £100 I65. 

112. 1 tun, at 25. 6d. per quart? Ans. £126. 

113. 1 tun, at 25 8d. per quart? Ans. £134 85. 

29. A nmnher of Articles. — Given the price of 1 
article in p(nice, to find that of any number — 

KuLE. — Divide the number b.y 12, for shillings and 



■1 I 



11! 



I< 



I i 




228 



PRACTICE. 



pence; and multiply the quotient by the number of 
ponce m the price. 

ExAMPLE.--WIiat is llio price of 438 articles, at 7d caol^ 
12)438 

365. Gd, the price at Id. each. 

20 )255"~6 

£12 15 6 the price at 7(/. each. 
438 articles at 1<Z. each will cost 438<;.=36s. Gt^. At 7d each 
2i¥l5" 6^ '' ''' much-or 7X3G.. Gd.JMs.'SuJ: 




EXERCISES. 

What is the price of 

114. 176 ib, at 3d. per lb .? Ans. £2 4s. 

115. 146 yards, at 9d. per yard ? Ans. £5 9s. 6d 

] ;^- ]^^ y^^^,^' '^* 101^. per yard ? Ans. £7 Us. 64 

117. 192 yards, at 7id. per yard ? Ans. £6. 

118. 240 yards, at 8^d. per yard ? Ans. £S 10s 

30. Wages.—mymg the wages per day, to find 
their amount per year — ° r j> 

RuLE.--Take so many pounds, half pounds, and 5 
pennies sterling, as there are pence per day. 

Example.— What are the yearly wages, at 5d. per day ? 

1 10 5 

5 the number of pence per day. 

7 12 ] the wages per year. 

£l^?nfS P^I'^''y;^« equal to 365rf.=240./.+120./.+r,J.= 
*i-f-lU* +6^/. Therefore any number of pence ner dav mimt 
be equal to £1 10.. 5d. multiplied by that number ^* 

What is the amount pei- year, at 

119. 3d. per day.? Ans. £4 Us. 3d. 

120. Id. per day.? Ans. iglO 12*. 11^. 

121. 9d. per day.? Ans. iE13 13*. 9^. 

122. 14^. per day.? Ans. £2\ 5*. IQd. 

123. 25. 6d. per day .? yl^w. ^'41 I*. 3^. 
124 ^d. per day .? Am. j(212 ISs. (]\d. 



pnACTfc:E. 



229 



BILLS OF PARCELS. 



Mr. John Day 



Dublin, IQth April, 1844. 
BouKlit of Richard Jones. 



(/. 



15 yards of lino broadcloth, at 13 per yard 

24 yanU of tsuperHtu) ditto, at 18 9 

!27 yards of yard ^vido ditto, at 8 4 

il) yards (;f drugg(it. at . 

1 2 yards of sorp;*', at 

?)'2 yards of shalloon, at . 



S 
2 10 

1 8 






£ s. 

10 2 
22 10 

11 5 
5 

1 14 

2 1?> 



d. 
() 




4 



Ans. jCSS 4 10 



Mr. James Paul, 



pair of worsted stockings, at 4 

(J pair of silk ditto, at . . ir) 

17 pair of tliread dllto, at . 5 

2'.) pair of cotton ditto, at . 4 

14 pair of yarn ditto, at . 2 

15 pair of Women's silk gloves, at 4 
I'J yards of Ihinnel, at . . 1 



Dublin, mMa\j,\UA. 

Bought of Thomas Norton, 
s. d. 

G per pair 

„ 

4 „ 
10 „ 

4 

2 

1}, per yard 






A,^s. £23 15 4J 



Mr. James Gorman, 



40 ells of dowlas, at 
;') 1 ells of diaper, at 
31 ells of Holland, at 
2'.> yards of Irish cloth, at 
\~\ yards of muslin, at 
13'j' yards of cambric, at 



s. 
1 
1 

5 



10 



Dublin, nth May, 1844. 
Bought of John Walsh & Co 

G per ell 

4^ „ 
8 „ 
4 per yard 
2\ 
(i 



[>'t yardsofpi'iMtod calico, at 1 2} 



5J 

11 



Ans. 



£h4 5 10] 



2'30 



l'n.\(TlVK 



mi 



■I h 



Liuly Denny, 



.'jjimlaofsilk, at . . 12 9 per yard 

]■> yards of floAvered do., at 15 G 

11-^yardsof Instring, at . 10 

J 4 yards of brccado, at . 11 3 

12] yards of satin, at . 10 H 

llg yard,i of velvet, at . 18 



Dublin, 20th Maij, 184-1. 
Bought of Iviuhard Mercer 






Mr. Jona^i Darling, 



5; 



15' lb of currants, at 

1 7| lb of IMalaga raisins, at 

l^i* lb of raisins of the sun, at 

17 lb of riee, at 

SI ib of j)Opper, at . . . x k> 
;> loa\ OS ol'sugar, weight 32,1 lb. at 81 

i:.'. oz. of cloves, at / . " '. D"per"oz 



Ans. £U 15 10 
JDublin, 2lst May, 1844. 

Bought of William Roper. 
•s\ d. 
4 per lb 



0" 

'6i 

1 6 



n 



Aus. £3 13 OJ 

• Mr. Thomas VVnght, -"""'"' '' "* ^""'' ^'^^^ 

Bought of Stephen Brown & Co. 

252 gallons of prime wliiskey, at 4 per gallon 
2r)2 gallons of old malt, at .08 
252gallonsof old malt, at . 8 



!) 



Ans. i:204 12 

MISCELLANEOUS EXEUCrSES. 

What is the price of 

i4 ISa". '2^u. 

2. a.>4 lb, at ]\d. per ib .? Ans. £1 ](]s. IQid 
Jl. 47o6 lb of sugar, at I2}d. per ib .^ Ans. £242 
ins. Id. 



.10 



£127 \Qs. 



25 pair of silk stoc 



kings, at C>s. per pair ? A 



'tis 



rUACJ'UK. 



231 



Ans. 

i. 
£242 

Am 



5. 3751 pair of gloves, at 2^. Cxi. ? Aois. £469 55 

6. 3520 pair of gloves, at 3a-. Gd. ? Ans £(516. 

7. 7341 cwt., at £2 C)s. per cwt. ? Ans. £16884 Gs. 

8. 435 cwt. at £2 7s. per cwt. f Ans. £1022 5s. 

9. 4514 cwt., at £2 lis. l},d. per cwt. } Ans. 
£13005 lO.v. ?,d. 

10. 3749f cwt., at £3 15a-. ChI. per cwt. } Am. 
£14153 17i-. iJJr/. 

11. 17 cwt., 1 (jr., 17 lb, at £1 4i-. 9^Z. per cwt. .? 
£21 lOi'. Sid. 

12. 78 cwt., 3 qr.s., 12 lb, at £2 17.?. M. per cwt. .? 
Ans. £227 lis. 

13. 5 oz., G dwt., 17 grs., at 5^. lOrZ. per oz. ? Ans 
£1 11. S-. 11,/. 

14. 4 yards, 2 (|rs., 3 nails, at £l 2.?. 4d. per yard .' 
A:ns. £5 4s. S^d. 

15. 32 acres, 1 rood, 14 perches, at £1 16^. per 
acre .^ Ans. £5S 4.9. If^Z. 

16. 3 ir:dlons, 5 pints, at 7a'. 6^/. per gallon.' Ans, 
£1 7,v. 21 d. 

17. 20 tons, 19 cwt., 3 qrs., 27} ft), at £10 10^ 
per ton .? Aius. £220 \)s. 1 1^^/. nearly. 

18. 219 toas, iij cwt., 3 qrs., at £11 75. Gd. per 
ton r Ans. £2500 13a-. O-^r/. 

QUESTIOXS IN PRACTICE. 

1. Wluit is practice .' [I]. 

2. Vriiy is it so called .' [1]. 

3. What is the dilforeneo between ali(|uot, and aliqnant 
parts.' [2J. 

4. Ihm are the ali(|Uot parts of abstract, and of 
applieato mniibers found } [3]. 

5. AVHiat is the difFoiencc between prime, and coni- 
pouiid aliipiot parts t [3j. 

fi. [fow is the pri(!e of any denomination fnmd, tli;-it 
of another beins given ? [(i and .8]. 

7. llrtn"- is the -prho of two or more denoi:iination3 
found, that of one bein;z uivi-n ? [7 and 9j. 

S. The p;-icG of ou:.> dc^nominatirM! being given, how 
do wo find that of any number of another .- [ lOj. 



I 






r H 



232 



PAACTICiC 



r. ^; F''?.,v '^ ^!''''" ^^ ^''y tlonomination is the aliquot 
part ot a shilling, how in the price of any number of that 
tlononuimtion louud ? [llj. 

10 AVhen the price of any denoniiuation is the 
ahquot part of a pound, how is the price of any num- 
ber ot that denomination found ? [12]. 

ri3 I ' ^^^'^' ^^ ^"^^"^ ^^ ^^'"^ complement of the price r 

li. When the complement of the price of any deno- 
minaum ih the nli^uot part of a pound or shilling, 
but the price IS not so, how is the price of any number 
ot that denomination found ? [13 1. 

13. When neither the price of a given denomination, 
1 or Its complement, is the aliquot part of a pound or 
M.iUmg, how do we find the price of any number of 
that denomian'ion ? [14, 15, lb', and 17] . 

14 How do we find the price of any number of 
articles whon t.^e pri(,c of each is an even or odd num- 
ber ot slullrngs, and less than 20 ? [IS and 19]. 

lo How is tlio price of a (pumtity, represented by a 
Tiaxod number, found ? [20] . ^ 

1<; How do we find the price of cwt., qrs., and lb, 
when the price of 1 cwt. is -iven .' [21]. 

17. How do we fin.l the price of cwt., qrs., and ib, 
when th.^pvieoof 1 11, isgiven.^ [22]. 

18. h\ vi' is the price of a ton found, when the prico 
of 1 In IS given ? [23]. ^ 

10. How do wo hud the price of oz., dwt., and grs. 
wh?.i the price 5f an ounce is given } [24]. ' 

^20. How do we find the price of yards, qrs., and nails, 
wucii tlie price of a yard is given .' [25]. 

21 How do we lind the price of acre^, roods, and 
porches .-' [PH]. 

22. How may the price of a hhd. or a tun be found, 
wlion the T;,ri'jo of a <piart is givcm ^ [27 and 28]. 
^ 23 Hov; rnny tlie price of any number of articles bo' 
round, the price of each in pence bciut^ given ? [29] 

. ^^^" ,^-i*''\'^''« ^^'=^K«« per year found, those - ■ .• day beino 
given ■' [30] ^ ^ * 



TARE AND TRET. 



233 



»ffp m 



■15^ 



TARE AND TRET. 

3'. The ^jrross \vei<^ht is tlio wei;;lit both of the 
goo('h, uii«,l oitho bag, &c., in which thoy arc. 

Tare Ih an jiHowanco for the bag, &c., wliich contains 
th(! aiti('h\ 

Suftle is the wci«;ht which renmins, after deducting 
the tare. 

Tret is, uynally, an allowaneo of 4 Vb in every 104 ft), 
or j'jj- of the weight of goods liable to waste, after the 
U\V{>. has been d^'dueted. 

Cioff is an allowance of 2 To in every 3 cwt., after 
both tare and tret \m\ti been deducted. 

What reuKiius after makin!? all deductions is called 
the vd, or nc/il wciu;ht. 

Diffjrent allowances are made in dilTercnt places, 
and for different goods ; but the mode of procetMlin;^ is 
in ^v^l ca^ii'S very simple, and may be understood from 
the ibllowin<ic — 



EXERCISES. 

1, I>ouo'ht 100 carcasses of beef at 18;$. ikl. per cwt.; 
gross wciglit 450 cwt., 2 qrs., 23 lb ; tret 8 lb per car- 
cass. What is to be paid for them ? 

100 carcasses. 

8 lb per carcass 
■ — cwt. qrs. H:) 
Tret, on the entire, 800 ib=7 IG 



cwt. qvc. lb. 
Cross 450 2 23 
'J "ret 7 10 



443 



2 7 at 18s, (Jd. per cwt.=.£410 5s. lOJfL 

2. What is the price of 400 raw hides, at 195. 10//. 
per cwt. ; tlie f^ross weight bein;^ 300 cwt., 3 q'.'s., l.o 
11) ; and the tret 4 lb per hide ? Ans. £290 3.7. 2^d.^ 

3. If 1 cwt. of butter cost £3, wh:it will be the price 
of 250 firkins; gro.na weight 127 cwt., 2 qrs., 21 tb ; 
tare 11 lb per firkhi .? Aiis. £:^(}i) Ss. 0-id. ^ 

4. ^Vhat is the price of 8 cwt., 3 (|rs., 11 lb, at Ins. \ 
fuL per cwt., allowing the usual tret .'^ A its. JLJti 1 Ia\ 
lO^i/. 





234 



TAKR AND TRET. 



G. What U tho price of 8 cwt. 21 lb at lfi» 4i./ 
por cw . , a I.nvin. the visual fret ? ^L ^7 4/ 81./- 
(>. Jou.h ^.^ h .J,s. of tallow ; No. 1 wc-ighin. ] J tt 
qr, 11 11, tare 3 <|r.s, ^u lb ; and No. 2, II cwt nr ' 

t.u>;';rtect^;t;rtV^^^--^ ^vL^^S;; 

A. . . cwt. qrs. lb. 

Orosa M-eight of No. I, K) 1 n 

(ii-oss weight of No. 2, 11 17 

(JrosH weight, . 
iui-e, 

Snttlo, 

Tret 1 lb per cwt. 



cwt 



. 21 
. 1 


2 

3 


. 1!) 
. 


2 22 
19^-§ 



cwt. qrs. lb. 
i are 3 20 
TuroO 3 14 

r~3 G 



, i. X^5 "tI}'!/. '' ^ '^- The price, at 30,. po, 
prupo^ti::^'' thirthef tret n.ay be ibund by the following 



cwt. cwt. qrs. tb. 
1 : I'J 2 22 



1 



lb. 

1().'10 



7. What is tho price of 4 hhds. of coppnra.s • No 1 
WCM,.,,, gross 10 cwt..,2,p..,4 tb,tafo^^;'s. 4lh? 

1-^ cwt 1 rp-., tare 3 qrs. I4 lb ; No. 4, 11 cwt 2 
'I'-s M I,, tare 3 q.s. IS ib ; the' tret b in" 1 7b ' r 
<;-t.,-nKl the price 10. per cwt. .^ An! mIoiM. 

«. Wliat will 2 bags of merchandise corao to • No 1 
w,Mg]in,g gross 2 cwt., 3 qrs., 10 lb- No o 'q .J' 

and at l.v. br/. per lb .? yj^^. £59 2* 8V 

^. A merchant has sold 3 bags of pepper • No 1 
Weighmo- m-o.SS 3 nwt 9--M.a . V. °o . i'\IW5 ^^^-Ji 



No. 3, 
1 It 



cwt. 2 qrs. ; No. 2, 4 



do th 



.> cwt., 3 qrs., 21 lb; tare 40 1b 
•per cwt. ; and the price boin'^ lor/ 



cwt., 1 qr., 7 lb 



y come to ? Ans. £74 1.?. 73.3^ 



por bag; tret 
per lb. What 



1 



10. liought 3 packs of wool 



<F., 12 lb; JN'o. 2,3 



weighing. No. 1 , 3 cwt. 



'-■? qis., ].-) ib ; tare 30 lb 



cwt., 3 qrs., 7 Ib; No. 3, 3 cwt 



"'> Rtoue; and at 10.?. 3^ 



per pack ; tret 8 lb f( 



amount to ^ 



per stone. Wliat do tl 



or f'v.'M-v 

- ■ --J 

Je» 



iAui: ANij via: I. 



835 



Vo. 1, 

No. 2, 
No. 3, 



owt. qrs. lb. 
3 1 12 
3 3 7 
3 2 15 



tb. 
Tare 30 
'J'aro 30 
Taro 30 



IJross, 10 3 6 
'I'aro, 3 6 


St. 

1 


90—3 qrs. G lb. 


Suttlo, 10 0=70 stones. 

st. st. lb. lb. 

20 : 70 :: 8 : 28 = 
st. lb. 
Suttlo, 70 
Tret, 1 12 


lb. 
J2 



Not weight, 08 4, at lO.-*. Gd. por stono=<£35 16s. 7}^d. 

1 1 . Sold 4 packs of wool at 9^. ^Jd. por stono ; woi<^h- 
ing, No. 1, 3 cwt., 3 qrs., 27 lb. ; No. 2, 3 cwt., 2 qrs., 
10 lb. ; No. 3, 4 cwt., 1 qv., 10 lb. ; No. 4, 4 cwt., 
jjr., G lb : tato 30 lb per pack, ami trot 8 It) for every 
20 stono. What is the price > Ans. £49 lbs. 2-^^%d. 

12. Bought 6 packs of wool ; weighing, No. 1, 4 cwt., 

2 qrs., 15 lb ; No. 2, 4 cwt., 2 qrs. ; No. 3, 3 cwt., 

3 qrs., 21 lb ; No. 4, 3 cwt., 3 qrs., 14 lb ; No. 5, 4 
cwt., qr., 14 lb : tare 28 lb per pack ; tret K lb for 
every 20 stone ; and at lis. 6d. per stone. What in 
the price ? Ans. £11 15s. Sfjfc?. 

13. Sold 3 packs of wool ; weighing gross, No. 1, 3 
cwt., 1 qr., 27 ft) ; No. 2, 3 cwt., 2 qrs., 16 lb ; No. 3, 

4 cwt., qr., 21 lb : tare 29 lb per pack ; tret 8 lb for 
every 20 stone ; and at lis. Id. per stone. What is the 
price } Ans. £4\ 13s. l\\\d. 

14. Bought 50 casks of butter, weighing gross, 202 
cwt., 3 qrs., 14 lb ; tare 20 lb per cwt. What is the 
net weight } 

cwt. qrs. lb. 
Gross weight, 202 3 14 
Tare, . .30 



qrs. cwt. 

O I 

iZ? 

14 = i 



cwt. 

202 


qrs. 
o 


tt). 

14 


20 






4040 lb. 




10 






5 


1 

o 

T 

2 


of 
of 



25| 



Net weight, 166 2 161 

i of the last, \ =the tare on 3 qr. 14 lb. 
2i = Xof thelast, 



Tare, 4057^ lb = 30 cwt., qr., 2-51 lb. 



I' 




236 



TARE AND TRET, 



lo Ihc gross weight of ton liluls. of tallow is 104 
cwt Z qrs., 2b lb ; and the tare 14 lb per cwt. WH-U 
IS the net weight ? Ans. 91 cwt., 2 qrs., 14-1 lb 
«J^"/^^ gi-oss weight of six butts of currants is 58 
ewt., 1 qr., 8 lb ; and the tare 16 lb per cwt. What is 
the net weiglit .? Ans. 50 cwt., qr., 1^ ft, 

17. What is the net weight of 39 cwt!, 3 qrs., 21 lb • 
the tare benig 18 lb pe. cwt. ; the tret 4 lb for 101 lb • 
and the cloff 2 \\, for every 3 cwt. > " ^^ ^^ l il> , 

cwt. qrs. lb. 
(xross weight, 39 3 21 
Tare, 



lb. lb. cwt. 

18= ! 1^=' 
^^- ^ 2=1^8 



cwt. qrs. lb. 
39 3 21 

5 2 23 
2 24 



1 13 



Suttle, . 
Tret=2'gtli, or 



Tare, 6 1 13 
2 lb m 3 cwt. is the .-^,th part of 3 cwt. i,- u 

Hence the cloff of 32 cwt. 26 lb is its ^^,th part, or 



33 2 
1 1 

32 2< 



2 

4 



O > 



Net weight, 32 4 

18._ What is the net weiglit of 45 hhds. of toliaeco • 

weighing gross 224 cwt., 3 qrs., 20 lb ; tare 25 cwt' 

3 qrs. ; tret 4 lb per 101 ; cIoiT2 lb for every 3 cwt ? 

Am. 190 cwt., 1 qr., 14^^ lb. ^ 

19 What is the net "weight of 7 hhds. of sn^-nr 
weighing gross, 47 cwt., 2 qrs., 4 lb ; tare in the whol..' 
10 cwt., 2 qrs., 14 lb ; and tret 4 ib per 104 \h > An^ 
3o cwt., 1 qr., 27 lb. ^ ' ' 

20. In 17 cwt., qr.. 17 lb, gross weight of V^alls 
how much net ; allowing 18 lb per cwt. ta?e ; 4 lb per' 
104 lb tret ; and 2 \h per 3 cwt. cloff? Ans. 13 cwt., 
6 qrs., 1 lb nearly. ' 



QUESTIONS. 

1. What is the gross weio-ht ? fSll 

2. What is tare? [31]. ^ ^ "'' 

3. What is suttle .? [31 ] . 
4 What is tret.? [311 ' 

5. What is cloff.? [31]. 

6. "Wliat is the iit-t weight.? [31]. 

7. Are the allowances made, always the same ? [31], 



qrs 


lb. 


3 


21 


1 


13 


o 


i> 




Ji 


1 


4 





20 





O > 



Am. 



237 



. i; 



SECTION VII. 

INTEREST, &c. 



1. Interest is the price which is allowed for the nso of 
money ; it depends on the plenty or scarcity of the latter, 
and the risk which is run in lending it. 

Interest is either simple or compound. It is simf^h 
when the interest due is not added to the sum lent, ^^, 
as to bear interest. 

It is compound when, after certain periods, it; is made 
to bear interest— being added to the sum, and considered 
as a part of it. 

The money lent is called the principal. The sura 
allowed for each hundred pounds " per annum" (for a 
year) is called the " rate per cent." — (per iilOO.) The 
amount ia the sum of the principal and the interest due. 

SIMPLE INTEREST. 

2. To find the interest, at any rate per cent., on any 
Bum, for one year — 

lluLE I. — Multiply the sum by the rate per cent., 
and divide the product by 100. 

Example.— What is the interest of £072 14s. U. for cno 
year, at 6 per cent. (XG fur every £100.) 
£> s. (J. 
672 14 3 




40-30 
20 



5 G 



7-25 The quotient, £40 7s. 3f/., is the iLtereet required. 



'J.Ort 



J lii 



We h 



ave divided by 100, by merely altering the decimal 




point [Se<3. I. 34J 



O' 



kS 



IN CERKST. 



II till- Interest w'.'i'c 1 percent,., it woulrl be the Imiidredth 
part of the principal— or the pj-ineip;il multiplied l)y ^^-f, ; but 
being b per cent., it is t> times iid much— or tlie principal mul- 
tiplied by y|^. 

3. Rule II. — Divide the interest into parts of cClOO; 
and take corresponding parts of the principal. 

EsAMPLK.— What is the interest of £32 4s. 2d., at G per 
cent. '. ^ 

£G = £5+£l,ov£^^lAns£^^-^5. Therefore tho in- 
terest is the J^ of the principal, plus the I of the J-. 



£ 
20)32 



s. d. 
4 2 



5) 1 12 2r} is the interest, at 5 per cent. 
6 5]- is the interest, at I per cent. 

And 1 18 7f is the interest, at 6 (5+1) per cent. 

'* EXERCISES. 

1. What is the interest of ^£344 lis. Qd. for one year, 
at per cent. > Ans. £20 1 3*-. 1 0\d. 

2. What is the interest of .£600 for one year, at 5 per 
cent. ? Am. i230. 

3. What is the interest of dE480 15.y. for one year, at 
7 per cent, t Jlns. ii33 135. 0|rZ. 

4. What is the interest of ^£240 10s. for one year, at 
4 per cent. > Am. £9 12s. Md. 

_ 4. To find the interest when the rate per cent, con- 
sists of more than one denomination — 

^ lluLE. — Find the interest at the higliest denomina- 
tion ; and take parts of tliis, for those which are lower. 
The sum of the results will be the interest, at the given 
rate. 

Example. —What is the interest of £97 8s. 4d., f(y one 
year, at £o lO.s. per annum '? 

£5 = £yi,0; and 10.'. = £,5,. 
£ s. d. '^ 

20)97 8 4 

10)4 17 5 is tho interest, at 5 per cent. 
9 9 is the interest, at lOs. per cent. 

And 5 7 2 is the interest, at £5-{-10s. per cent. 



£5 






INTEKEST. 



239 



At 5 per cent, tho intevcst is the .1,- of fhe piMneipal ; at 
lOs. per c.-nt. it is the j\ of wluit it U at 5 |.er cent. There- 
fore, at £5 lOi'. per cent., it is the sum of huth. 

5. What is tho interest of /J371 IDs. ly. for one 
year, at i:'3 155. per cent. ? Am. jSIS 18.v."llf^. 

ei. AVhat is the interest of i^84 ll.y. lOirZ. for one 
year, at £4 5.v. per cent. } Ans. j£;3 II.9. lOfr/. 

7. What is the interest of JCOI O5. 3|<Z. for one year, 
at £6 12.9. 9^/. per cent. } Am. M O5. \0\d. 

8. What is the interest of £ms bs. for one year, at 
£b 14a-. i)d. per cent. } Am. £bb 8a-. 8r/. 

5. To find the interest of any sum, for several 
years — 

Eui.E. — Multiply the interest of one year by the num- 
ber of years. 

Example.— AVJiat is the interest of £32 145. 2(/. for 7 
years, at 5 per cent. '? 

£ A-. d. 
20)32_14_2_ 

i 12 81^ is the interest for one year, at 5 per cent. 



And 1.1 S 11^ is the interest for 7 years, at 5 per cent. 
This rule requires no explanation. 

EXEilCISKS. 

9. Vvniat is the interest of £U 2s. for 3 years, at (3 
per cent, r Ans. £2 U)s. dd. 

10. What is the interest of ,£72 for 13 years, at m 
10a-. per cent. } Ans. £m IG.v. 9*^/. 

11. What is tho interest of £Sb3 Qs. i)\d. for 11 
years, at £4: V2s. per cent. } Ans. £431 vSs. l^d. 

().. To find the interest of a given sum for years, 
niontlis, &c. — 

lluu:. — Having found the interest for the years, as 
ah-eady directed [-2, &c.], take parts of the interest 
"jr that of tho mouths, &( 



the results. 



and tl'sn add 



<U 



! I 




'■rt. 



l^ui 



I j-,jj i 



Hi 






^•40 



INTEREST 



Example —What is the iz^tercs,* of J£86 85. M. for 7 vcars 
and 5 months, at 5 per cont. > ^ 

20)86 8 4 



4 G 5 13 the interest b> : T?a>, ai 5 jxv • cep 



1 « o.~^"" A S *if ?^ *^'® interest for 4 monthv 
1 8 9^-^4 =0 7 21 IS the interest for 1 month. 

And 32 llj is the required interest. 

EXERCISES. 

12. What is the interest of ^£211 5^. for 1 year and 
6 months, at 6 per cent. .? yl-^w. ^19 0.9. 3d 

13 AVhat is the interest of £514 for 1 year and 7-i 
months, at 8 per cent. > Ans. £66 16^ 4U ' 

1-1. What is the interest of £1090 for l' year and 5 
months, at 6 per cont. } Ans. £92 I3s 

1;^ What is the interest of £175 lO.^. 6^. for 1 year 
and J monriis, at 6 per cent. .? Ans. £16 135. 5//^.^ 

o. A\Iuit IS the interest of £571 15.. for 4 years 
and 8 months, at 6 per cent. .? Ans. £160 1.. 9-^-./ 

17 VViiat IS the interest of £500 for 2 years and 10 
iiiwntlis, at 7 per cent. ? Ans. £99 3.y. Ad 

lb". What is the interest of £93 17.':. Ad. for 7 yc^a-s 
,}^T'^^'f' ^'^' ^ P^^' «^"t- • ^'I'i'^-. ^14 11.. lid " 

and ^ ^ ;f '•' /'-' ^"^''''^ '^^ ^'^-^ '^-^- ^'^^- f^^- 8 y^^^ 
and S mouths, at o per cent. > Ans. £36 ll.v. 111,/. 

O) or b, &c., per cent. ' 

At 5 per cent. — * 
■ KuLE.— Consider the years as sh:lHn..s, and the 
montlis as pence ; and find what ali-pot pa^ or part 
^fS^^^n^r^^"^- ^^-*^^-^--4^tor^a.. 

To find the interest at 6 per cent., find the interest 
at 5 per cent., mid to it add its fifth part, &c. 
Ihe mtcrest at 4 per cent, will bo the 



per cent minus its fifth part, &c. 



interest at 



INTKRE6T. 



2^1 



d. for 7 years 



4^ 5 p<\ • cep 



s. 
bh. 



1 year and 

ear and 7^^ 

yx^ar and 5 

for 1 year 

''• ^1 0*^- 

3r 4 years 
s. 9^(1, 
ars and 10 

■or 7 years 
Is. l\d. 
n- 8 years 
v. 111^/. 

y time, at 



and tliG 
t or parts 
t or parts 

LJ interest 

iterest a^ 



^ 8. Example L— What is the interest of X427 5^. 9c/. for 
years jind 4 luuiiths, at 5 per cent. '\ 

years and 4 months are represented by 6s. 4r/. : but 
0.. 4,/.=...s..+],.+W.==<-}-J^ of a pou)td + the i of ths J,. 

4)427_ .5__0_ 

5)100 To ,5y is tlie I of principal. 
;])21 7 31 is the ^V(l of]) of principal. 
_-!-._r_A ''^ ^^'"^ s'" <^'^ ''^' ^'') ^''" Pi'i'^^ip'd. 
And l;>.5 G 1^ is the required interest. 
The intcres^t of £1 for 1 year, at 5 per cent., would be 1.. 
lor 1 inouth 1</. ; for any number of years, the same nuinl)or 
ot shillnigs; for any number of nioiitlis, the same number of 
pence ; and for years and months, a corresponding nund>er of 
shiUmgs and pence. But whatever part, or parts, these sliil- 
xHigs and i>ence are of a pound, the interest of any other sum 
tor tJie sauio time and rate, must be the same part or parts of 
tJiat otlier sum-since the interest of any sum is proportional 
to the interest of JSI. 

KxAMPi-E 2.— AVhat is the interest of £14 2s. 2d. for G 
years and 8 months, at G per cent. ? 

G.'v. 8^/. is the .' of a pound. 

£ s. d.' 
3)14_2^2_ 

5)4 14 {)} is tlio interest, at 5 per cent. 
18 W'l is the interest, at 1 per cent. 

5 12 KJi is the interest, at G (5-f-l) per cent. 

EXERCISES. 

20. Find the interest of .£1090 17.?. Qd. for 1 year 
and 8 months, at 5 per cent. .? Am. £90 18.v. \\d. 

21. Find the interest of £976 14,?. Id. for 2%^ears 
and 6 months, at 5 per cent. } Am. £122 \s. 9|rZ. 

22. Find the interest of £780 17s. fi^i. for 3 years 
and 4 iiiontlis, ut G per cent. > Am. £1.^)6 'M. 6d. 

23. What is tlie interest of £197 lis. for 2 years 
and 6 montli.s, at 5 per cent. ? Am\ £24 13.?. l{)ld. 

24. What is the interest of £279 lis. for 74- months, 
at 4 per cent. .- A as. £6 19s. 9,-^,//. 

7U( 



Jit; 



2.1. What IS iho. iuterest of £790 IGs. for 6 
and S months, at 5 per cent. > A'us. £263 12j. 



year 



-^m 



y 



212 



INTKREST, 



26 What is the interest of ^^121 2s. \U. for 3 years 
and .3 inontiis, at 5 per cent. ? Ans. £oq 3^ 53,^ "^ 

27 ^VliMt i,s the interest of i2l837 4*. 2d. foVs'vca.-a 
and K) inontlis, at 8 per cent. } Am. £563 S.y. 3d. 

9. When the rnU^ or nuin])er of years, or loth of 
thein, are expressed by a mixed nuinber- 

l.ULE.— Find the interest for 1 year, at 1 per cent , 
and multiply this by the number of pounds and the frac- 
tjun of a pound (if there is one) per cent. ; the Mini uf 
these products, or one of them, if there is but one, will 
give the interest for one year. Multiply this bv' the 
number of years, and by the fraction of i year (if^lhe e 
f one) ;_ and the suni of tliese products, or one of them' 
If there is but one, will be the required interest 
atfpoTce'atT*"^ *'' interest of i:21 2.. 0./. fur 3| years 

£21 2s-. G./.^100=4.. 2-ld. Therefore 
.t s. d. 

4 25 is the interest fori year, at 1 per cent. 



lit Jo. 



1 1 1' is the interest fori year, at 5 per cent. 

a 1 ? 1 A n^ the interest for 3 years, at do. 

i> lo luj IS the interest for 2 ofa year (£1 Is. l^^^.x"'),; 

3 19 31 is the interest for 3^- years, at do. 

J^^':.^''^ '' *^-i'^terestof £300fbr5.years, 
£300^100=3 LUhe interest fori year, at 1 per cent. 



9 OLs the interest fori year, at 3 per cont. 
__ ___^^ tlie interest for 1 year, at £■' (cC3x^) 

11 5 is the interest fori year, at 3| per cent. 

56 5 Is the interest for 5 years, at ^ por cent 

- 10 3 IS tlio do. for l- pru- (i:5 12^. G^r/.J-2A 



And 04 13 9 is the interest I) 



r^■■t 



or ir^ years, at Z} do. 



'or 3 ycarfl 
, 6^1. 
fur 3 ycai'3 
<s. 3d. 

•1* both oC 

por cent., 
(1 the fruc- 
he sum vi' 
t ono, Avill 
lis by the 
' (if (lierc 
B of them, 

!t. 

>r 3| years 



x;'),utJo. 



5^ years, 



per cent, 
oer cont. 
por cent. 
' peroenl 

t 3^ do. 



rXTERRST. 



EXRRCISKS. 



243 



28. What is tliG interest of JE379 2s. Gd. for 41- years, 
at 5 1 por cent. ? Aiis. £<ji 5.5. 5^/, ^ ^ » 

29. What is the interest of .£640 \Qs. Qd for 24 
years, at^4| per cent. > Am. £72 1,?. 2j\d. 

30. "What is the interest of £600 IO.9. Qd. for 3i 
years, at 5|- per cent. } Ans. £11.5 2^. QJ-fZ. '' 

31. What is the interest of £21? ^s^.\id for e^ 
years, at 5 5 per cent. .? Ans. £81 8,?. bl-d. "* 

10. To find the interest for days^ at 5 per cent — 
'/^.^^•— ^^"Itiply the principafby the number of days, 
and divide the product by 7300. 

E.VAMPLK.— Whatis the interst of i;2G 4.s. 2d. for 8 days? 

£> s. d. 

2G 4 2 

8 

201) 13 4 
20 



4193 
12 



r300)50320(6:!-|-'W. 
•43800 ' 



6520 

U ItVZT'"'] "l^T'*/.' .^'^« ^' ^' 7^— «ince tlie remainder 
IS gieatey than lialf the divisor. 

The interest of £1 for 1 year is £J^, and for 1 day Jj-f-3G5= 

20K3tl5="300; that is, the 7300th part of the principal. 

Therefore the iritorest of auy other sum for one day, is the 

.?00th part of that sum; and for any number of da^s. it a 

that number, multiplie.l by the TSOOtli* part of tl>e princ ipal- 

nunrbpl'of T "'' 'T^- '-^'/"f' '^'' ^'^""^^^^ multiplied bf he 
number of days, and divided by 7300. ^ 

EXKRCISES. 

33. Find the interest of £140 lO.f. for 76 days, at 5 
per cent. Ans. £1 9s. 3^%^jd. 

33. Find the interest of £300 for 91 days, at 5 per 
cent. '•'■'• ^"^ '-'- no., 7 -^ ' 1^^ 



Ans. £3 I4s. Q^d 



34. What is the interest of £800 fur 61 day 



i ; i 



,i' 



, ^ 1 



per cent. ? Am. £6 1.3.s-. S%%d 



s, at b 




241 



I.NTKKKST. 



11. To fmr] t],o intovost f<n- dnys, at .^;/7/ other rate— 



pitrls of llii 

KXAMI'MO 

fliiys, at cCG 

X3324 Gs. 

£ .V. 
5)5 
2)1 

OJU 

And G lo" 
je 1+105. 
This rule 



1 111,1 the mtercj.st at 5 per cent., and take 
■* tor the remainder. 

—What is the interest of Xr.324 C.<. 2<Z. for 11 
ivs. per cent. ? 

2^.Xll-^7;500=£5 0.. 2],/. Therefore 

2| is the interest for 11 days, at 5 per cent. 
J .J IS the interest for 11 days, at 1 per cent. 
JJ_ IS the interest for 11 days, at 10,v. per cent. 
2J is the interest for 11 days, at jGG 10*-. (£5-f 

requires no explanation. 

EXERCISRS. 

w ?f' ^u'^V" *^»e interest of £200 from the 7th May 
to .the 2bth September, at 8 per cent. } Ans. £Q As 

30. mat is the interest of ^£150 15^. Qd. for 53 
aays, at 7 per cent. } Am. £\ \0s. 7^d 

37. What is the interest of d8371 for l"year and 213 
days, at 6 per cent. .? Ans. i235 os. Qd 

A.!ll' .Tl''''^ "' ^\ ''l^'^'f^ °^ ^'-^^ ^'''' ^ y^-^'' and 135 
aays, at 7 per cent. } Ans. £23 0.s\ 3^^;?. 

Sometimes the number of days is"the\liquot part of 
a year ; m which case the process is rendered more easy 

l^^^-^:ti^^ '^''''' '' ^^'^ '- ' ^- -^ 

1 year and 7-3 days=] i year. Hence tlie required intercsit 
il75 '^r7'' ^'" 1 /f- +i.t« lifth part. ]^ut Ihe interest':? 
fjiio tor 1 year, at the given rate is £14. Tlierefore its 

£1g'ig1 gi^«^ti>^« i« X14+i:y=:£i4+i2Ta..i 

12. To find the interest for mon/Jis, at 6 per cent-^ 
±IULE.— if the number expressing tlie months is even, 
nmtoply the prmcipal by half i/>f number of month 
and dn,de by 100 But if it is odd, multiply by tho 
hali of om Jess than the number of montlis ; divide the 
result by 100 i and add to the quotient what will bo 
obtained if we divide it by one les than the nn.mIo of 



11 



INTEUIiST 



345 



! ti 



KxAMiT.K 1.— What is the interest of X72 (Js. id. for 8 
months, iit G per cent? 
£ ,s. (I. 
72 4 
4 



£2H0 5 4 
20 



17-85s. The required interest h £2 Us. lOUl. 
12 



10-24r/. 
4 



0-90= J f/. nearly. 
Solving the question Ity tiic rule of three, wo shall have — 
;eiOO : i:72 Gs. 4^/. : : JCG : £72 6.s-. 4f/.x8xG 

12 : 8 l(Xrxl2 =(^^ivul. 

ing Itoth numerator and denoniinator by G [Sec. IV. -' 
X72G.9. 4r/.x8xG-^G .£72 Gs. 4r/.xS 

100 x 12-^0 = lUO X 2 ~" = (dividing ooth 

numerator and donominator by 2) ' ' -" ■^^^yczo_J_o'~~ 

£12 G.;. Aily^A 

100" 

— that is, the required intore.sfc is equal to tlic given sum, 

nuiKiplied by half the number which expresses the montliH, 

and divided "by 100. 

Ex-AMPi.E 2.— AVhat is the interest of £84 Gs. 2d. for 11 
months, at G per cent. ? 11=10+1 10-^2=5. 

£ s. d. 
84 6 2 

•) One loss than the given number of 
n-iHol^ inonths=10. 

20 
£ s. d. 

4oOs. i0)4 4 Ojj- is the interest for 10 niontliii, at U ptT oc'iit. 
12 8 5| is the intore!.t lor 1 nujutl), ut Siinie rattj. 

3-70(7. And 4 12 9 is the iutciest lor IJ (ID-j-I) moutU-i, ut ti ;•* 

4 



2-80f.=.^J. nearly. 
Tlie interest 
11 -1 month, pUis the "udovost of II — I monlh -r-1 1 — 1- 



11 months is evidently the interest ivf 



I 



.M 



948 



INTEItnsT. 



ual > 



'IXKKCISES. 

39 What k tlio intorost of £250 17s 6d for <( 

«fr ,;,,.'"■ "!,"""■"■'"'''' ■^■5^1 15s. for 8 months. 
«*'. per (,..(.- ^«. Jtaa 17s. 4.id ' 

6 pt or„t.".' "i!;.: i^'^rres"' ^'''° '"' ' "■"'"■'^' ^' 

6 p1fr ofntt ^f i"r^' "' ■'"'' '" '" '™"*^' "' 

at P,i^ "^^^r^^^^t ''■ '- ' -""■»' 

...i' v^ir^*^ *''? '"''"'™' '« ?»■<! l>y -^nj/s, multinlv the 
sam by the nnmboi- of days which have elamcd b^fom 
any payment wa. made. ' Subtract Z trjfavu „, 
and multiply the remainder by the number of S 
wh.eh passed between the first and second paymon?, 

Ir r ^"'" "'T'^ P^>'™""' »■«' '""'"-ply «l.is nS: 
seconJ !nd "ir^'' "^ ''"^^ '^'™'' passL/ between the 

days more £20 : in 15 mor^ £^o "^ i f .f'*'"^^ ~^ '^ • ^" '^ 
day,, an, the ^SeTat"' Cf Jlrj^^e^'ireit 

X days. £ day. 
nZx 6= 702xi 1 
lOOx 7= 700x1 

i^0xl5=1200xlf=°C^^^<^- 

48x60=3168x1 J 

Wo?e''''* '^'^"' ^^^ 1 ^-y- -*S Ver cent., is 15.. 9|c/ 



INTERRST. 



24/ 



..... i 



fi)0 15 \)'^ is tho iutorost, at 5 pei cent. 

'^ 2^ is tho interest, at I per cent. 

Jl)0 18 11^' i8 tho intorest, at per cont. 

G 4 is tho interest, at 2 per cent. 

And 1 5 3J is tho interest, at 8 per cent., for th*^ given 

sums and iiiuos. 

If tho entire sum were G days unpaid, tho interest would be 
tho aafoe as that of 6 times as much, fbr 1 day. Next, £100 
due for 7 days, sliuuM proiluce as much as £700, for 1 day, 
&c. And all tlio sums du ■ for tho diffcrci>t periods should 
produce as much as the sum of their equivalents, in 1 day. 

EXERCISES. 

45. A merchant borrows i2250 at 8 per cent, for 2 
years, with condition to pay before that time as much 
of the principal as he pleases. At tho expiration of 9 
months he pays jL'80, and 6 months after £70 — leaving 
the remainder for the entire terra of 2 years. How 
much interest and principal has he to pay, at the end 
of that time > ins. J£;i27 16^. 

46. I borrow ^£300 at 6 per cent, for 18 months, 
with condition to pay as much of the principal before 
the time as T please. In 3 months I pay £G0 ; 4 months 
after i^lOO j and a months after that £75. How much 
principal and interest am I to pay, ut tho end of 18 
montiis.? Ans. £,"^9 los. 

47. A gives to B at interest on the 1st November, 
1804, £6000, at 4^ per cont. B is to repay him with 
interest, at the expiration of 2 years —having liberty to 
pay before that time as nmch of the principal as he 
pleases. Now B pays 

£ 

900 



The IGth December, 1804, 

The 11th March, 1805, 

The P.i'Ui March, 

The 17th August, 

The 12th February, 1806, 



1260v 
600 
800 

1048 



How much principal and interest is he to pay on th# 
1st Novembor, 1806 ? Ans. £1642 9^. 2if ||-fZ. 
48. Ticntat interest £600 the 13th May," 1833, for 




24n 



INTIRIST. 






1 year, at 5 per cent—with condition that the receiver 
may diHchargo as much of the principal before the tinm 
ns ho pk.as,vs. Now ho pays tlio i)tli July .£^200 ; and 
the 1/th beptember i^l50. How nmch principal and 
mtorost IS ho to pay at the expiration of the year ? 
Ans. £26Q 13s. b^\d. ^ 

I ^'^' }^}^yv^^ that the pupil, from what he hag 
learned of the properties of proportion, will easily un- 
derstand the modes in which the following rules are 
proved to be correct. 

Of the principal, amount, time, and rate— given any 
three, to find the fourth. ^ ^ 

Given the amount, rate of interest, and time : to find 
the prmcipal — ' 

^ Rule.— Say as £\Q0^ plus the interest of it, for the 
given tune, and at the given rate, is to ^100 ; so ia the 
given amount to the principal sought. 

^^ExAMPLE.-Whiit will produce £862 in 8 years, at 5 per 

giv1n'rfe.^'Tf2ir? "'""^ '""^ ^''' ^" ' ^^^ ^' *^« 

£140 : £100 : : £862 : J^^^^ =£615 14.. 31^. 

When the time and rate are given 

n.„^ll!? • ^^^ °^^''"' ^"""^ • ' ^"*®^^'** of £100 : interest of 
that oilier sum. 

By alteration [Sec. V. 20], this bocomes- 
th^^mn" '°*"''^* ""^ ^^^^^ '' ■ "^"^ °*^®'' '"'"' • ^°*^''^^*^ ^^ 

rw v%T-i"° "th« first + the second : the second," &c. 
L^ec. V. 2\)] we have — 

f^fi^^ i I*' interest : £100 : : any other sum -f- its in- 
terest . that sum— which is exactly the rule. 

EXERCISES. 

49. What principal put to interest for 5 years will 
nmount to £402 10.., at 3 per cent, per annum ? Ans, 



50. What principal put to interest for 9 years, at 4 
percent., will amount to £734 S.. .=^ Ans. 



INTEREST. 



249 



51. Tho amount of a certain principal, bearing inter- 
est for 7 y(;ars, at 5 per cent., u jL'334 16*. What is 
tho j)riucipul ? Ans. £24ii. 

1.'5. Given tho time, rato of interest, and principal— 
to find tho amount — 

lluLK.—Say, as JEIOO is to JCIOO plus its interest for 
the given time, and at the given rato, so is tho given 
sum to tho amount required. 

ExAMPLK.— What will £272 jomo to, in 5 years, at 5 per 

C6Ht* f 

^'fi}^r (='^1^0-f-£5x5) is the principal and interest of 
*.10U fur 5 years ; then — 

£100 : £125 : : £272 : ^~=.£ZiO, the required 

amount. 

We found by tho last rule that 

£iOO+its interest : £100 : : any other sumf its interest : 
that sum. 

Inversion [See. V. 20] changes lliis into, 
£100 : £100-f-its interest : : any other sum : that other 
Bum-fits mterest— which is the pi.vBont rule. 

EXERCISES. 

52. What will £350 amonnt to, in 5 years, at 3 per 
cent, per annum ? Ans. £402 10a\ 

53. What will £540 amount to, in 9 years, at 4 per 
cent, per annum ? Ans. £734 8.y. 

54. What will £248 amount to, in 7 years, at 5 per 
cent, per annum ? Ans. £334 16s. 

55. What will £973 4s. 2d. amount to, in 4 years 
and 8 months, at 6 per cent. ? Ans. £1245 145. l^d. 

56. What will £42 3^. Qid. amount to, in 5 years 
and 3 mouths, at 7 per cent. ? Ans. £57 13.?. lOirZ. 

16. Given the amount, principal, and rato — to find 
the time — 

Rule. — Say, as tho interest of tho given sum for 1 
year is to th • • • - 



juired time. 



given interest, so is 1 year to the re- 




250 



INTEREST. 



X14 l5. 8rf. (the interest of £281 13s. 4d. for 1 year r21) • 

£56 6s Sd. 



.^56 Gs. iM. (the given interest) 
lequircd number of years. 



1: 



X14 Is. 8(/.=^' *^« 



17. 



iience bij 



• + . r , .'^^'%> to find the time— Divide the 

InZlt '^l^'"^'" ^"'^'^^^^ ^^^ 1 3^^^^' i«to the entire 
interest, *ud the quotient will be the time. 

in/ln'f •'^'^^''*' *^'" P"°«ipal, and rate beina; c-iven, the 
ntere«t is prcpo-uonal to the time; the longer the Jme the 
more the interest, ^ud the reverse. That is- ' 

ihe interest for one time : the interest for another • • 
the former time : the latter. <iuouier . . 

^ Hence the interest of the given sum for one year rthe 
nterest for o,ic time) : the given interest (the interest of 
the same sum for amther time^ • • 1 vpo^ h\Z ,. ^^^^^f.^* 
produeedthe former) : thHfi- sought tiatXiX^^^^^^^ 
uuced the latter)-.which is the rule. ^ 



EXERCISES. 

57. la what time wciild .^300 amount to ^£372, at 6 
per cent. > Ans. 4 years, ' 

58 In what time would £211 5s. amount to ^£230 
moniht' ^'' ''''*•• ^^'- ^" ^ y^^^ ^^^ 6 

59. When would £561 15s. become £719 Os 95^ 
at 6 per cent. ? Ans. In 4 years and 8 months. * ' 

60. When would £500 become £599 3s. 4d., at 7 per 

""^^V .,r<^'""- ^^ ^ y^^^"« ^n^ 10 months. ^ 

61. When will £436 9s. 4d. become £571 8s Ud 
at 7 per cent. ? Ans. In 4 years and 5 months. ' ' ' 

the rat^""' *^'^ amount, principal, and time— to find 

EuLE— Say, as the principal is to £100, so is the 

given interest to the interest of £100— which will give 

he interest of £100 at the same rate, and for the same 

he rat ' ' ^ *"''''' ^""^ *^^' ^''^*^'^* ^" ^« 



INTEREST. 



251 



Example.— At what rate will £350 amount to £402 10s 
in 5 years ? 

£350 : £100 : : £52 10s. : ^^^ 10.-. x 100 



350 



=£15, the in 

terest of £100 for the same time, and at the same rate 
IJien 'j=3, is tbi required number of years. 
We have seen [14] that the time and rate being uie same, 

£100 : any other sum : : the interest of £100 : interest 
of the other sum. 

This becomes, by inversion [Sec. V. 29] — 

Any sum : £100 : : interest of the former : interest of 
100 (for same number of years) . 

But the interest of £100 divided by the number of years 
wliich produced it, gives the interest of £100 for 1 year— 
or, in other words, tlie rate. 

EXERCISES. 

62. At what rato will c£300 amount in 4 years to 
i£372 ? Ans. tj per cent. 

63. At what rate will £248 amount in 7 years to 
£334 16s. ? Ans. 5 per cent. 

64. At what rate will £976 145. 7d. amount in 2 years 
and 6 months to £1098 IQs. 4^d. } Ans. 5 per cent. 

Deducting the 5th part of*the interest, will give the in- 
terest of £070 145. Id. for 2 years. 

65. At what rate will £780 175. Gd. become £937 
Is. in 3 year.1 and 4 months ? Ans. 6 per cent. 

66. At what rate will £843 5^. 9d. become £1047 Is. 
7|<-7., in 4 years and 10 months ? Ans. At 5 per cent. 

67. ^ At what rate will £43 25. 4JyZ. become £00 75 
4J-rf., in 6 years and 8 months } Ans. At 6 per cent. 

68. At what rate will £473 become £900 135. fii^Z 
in 12 years and 11 months ? Ans. At 7 per cent. 



COMPOUND INTEREST. 



19. Given the principal, rate, and. time — to find the 
amount and interest — 

lluLE I. — Find the interest due at the first time of 
payment, and add it to the principal. Find the interest 



■t 


H 




r 


i 1 


1 


• t 




1 , 


1 1 


I ■ 


m 1 




■■ 




;; i 


■ 




1 




1; , 




K i. :. J 


. ; , 



ilHiiJ 



■i m 



m 



252 



INTEREST. 



of that sum, consiJorotl as a new priucipal, and add it 
to what it would produce at the next payment. Con- 
sider that new sum as a principal, itud proceed as 
before. Continue this pi-ocess through all the times of 
payment. 

Example.— What is the compound interest of £97, for 4 
years, at 4 per cent, lialf-yoarly '? 
£ s. d. 
97 
^ 3 17 7.i is the interest, at the end of 1st half year. 

100 17 1\ is the amount, at end of Isfc half-year. 
4 8^- is the interest, at the end of 1st year. 

104 18 3'/ is the amount, at the end of 1st year. 
4 3 11| is the interest, at the end of 3rd half- year. 



109 
4 



2 
7 



3 is the amount, at the end of 3rd half-year. 
3L is the interest, at the end of 2nd year. 



113 
4 



118 
4 14 



9 0.} is the amount, at the end of 2nd year. 
10 9.^ is the interest, at the end of 5th half-vcar. 

4 is the amount, at the end of 5tli luilf-year. 

5 is the interest, at the end of 3rd year. 



122 14 9 is the amount, at the end of 3rd year. 

4 18 2\ is the interest,* at the end of 7th half-year, 

127 12 Hi is the amount, at the end of 7th half-year, 

5 2 1^ is the interest, at the end of 4th year. 

132 15 0;i is the amount, at the etui of ith year. 
97 is the principal. 

And 35 15 0^- is the compound interest of £97, in 4 years. 

20. This is a tedious mode of proceeding, particularly 
when the times of payment are numerous ; it is, tlun-e- 
foro, better to use the following rules, which will be 
found to produce the same result — 

KuLE II. — Find the interest of £1 for one of the 
payments at the given rate. Find the product of so 
luany factors (cdch of them c-Gl-fits interest for o!ie 
payment) as there arc times of payment ; multiply this 
product by the given principal ; and the result v/ill bo 
the principal, plus its compound interest for the given 



Ur 



INTEREST. 



253 



time. From this subtract the principal, and the remain- 
der will be its compound interest. 

Example 1. — What is the compound interest of £237 for 
3 years, at 6 per cent. '? 

£0Q is the interest of £1 for 1 year, at the given rate ; 
and there are 3 payments. Therefore £1-06 (^£l-\-£0-\j) ig 
to be taken 3 times to form a product. Hence lOGxlOOx 
l-06x£237 is the amount at the end of three years; and 
l-0Gxl06xl-06x£237— £237 is the compound interest. 

The following is the process in full — 

£ 

1-06 the amount of £1, in one year. 
1'06 the multiplier. 

11236 the |j,mount of £1, in two years 
lOG the multiplier. 



1-191016 the amount of £1, in three years 
Multiplying by 237, the principal, 

£ s. d. 

wo find that 282-270792=282 5 5 is the amount • 
and subtracting 237 0, the principal, 

we obtain 45 5 5 as the compound interest>. 

Example 2. — What are the amount and compound inte- 
rest of £79 for 6 years, at 5 per cent. 1 

The amount of £1 for 1 year, at this rate would be £10.5. 
^ Therefore £1-05 X 105 X 1-05 X 1 '05 X 105 x 1-05x79 is the 
amount. &c. And the process in full will be — 

£ 

1-05 
105 



11025 the amount of £1, in two years. 
11025 



1-1^1551 the amount of £1, in four years. 
1-1025 



1-34010 the amount of £1, in six years. 
£ s. a. 



£105-86790=105 17 4| is the required amount. 
79 



And 26 17 4| is the rcqnired interest 

M 2 





2.11 



tNTEKKST 



Example o. — Whataro the aiTJount, and compound interest 
of £27, for 4 years, at £2 10s. per cent. Iialf-yearly. 

The anionnt of XI for ono pnyinont is X102:>. Therpfore 
ClO-iry X I- 025 X 1 025 X 102o x 1025 x 1-025 x 1025 x 
I 025 X27 is the amount, &o. And the process in full will be 
£ 
1025 
1025 



1 05003 the amount of XL in one year, 
105003 "^ 



T- 10382 the amount of XI, in two years. 
1-10382 ■ 



1-21842 the amount of XI, in four years. 

27 , ^ 
£ s. (17 



X32-8U734=32 17 11| is the required amount. 
: 27 



And 5 17 li| is the required interest. 

21. Rule IIT.— Find by the interest table (at the end 
of the treatise) the amount oi £,\ at the .f^iven rate, and 
for the given number of payments ; multiply this by tho 
given principal, and the product will be the required 
amount. Prom this product subtract the principal, and 
tho remaiader will be the required compound interest. 

Example.— AVhat is the amount and compound interest 
of X47 lOi. for 6 years, at 3 per cent., half-yearly '? < 

X47 10,y.=X47-5. 
We find by the table that 

X1-4257G is the amount of XI, for the given time and rate. 
47-5 is the ■ .altiplicr. 

~rr~~ ^ •"'• ''• 

67-7230=67 14 5'' is the required amount. 
47 10 



And 20 4 5^ is tho required interest. 

22. Tlule r. requires no explanation. 

li,KAso\' OK F.uLK II. — Whon the time and rate are Mie 
Bfvme, ivfo priicipnls are proportional to their corresponding 
amounts. Thcrofore 

£1 (one principal) : £1 03 (its corresponding amount) : 
£10G (.another principal) : £1-00 X lOG (its corresponding 
amount). 



IXTF.UKST. 



2i?5 



Ileneo the amount of £1 for two ycirs, is £106xl'06— 
or the product of two factors, each of them the amount of £1 
for one yeur. 

Again, for similar reasons, 

£1 : £1-06 :: £l-06Xl-0G : £1 -OGXl-OGXl-OG- 
Hence the amount of £1 for three years, la £l'06xl'06Xl*06— 
or the product of three factors, each of them the amount of 
£1 for one year. 

The same reasoning would answer for any number of pa^'- 
ments. 

The amount of any principal will be as much greater than 
the amount of £1, at the same rate, and for the same time, as 
the principal itself is greater than £1. Hence we multiply 
the amount of £1, by the given principal. 

Rule III. requires no explanation. 

23, When the decimals bejonie numerous, we may 
proceed as already directed [Sec. II. 58]. 

We may also shorten the process, in many cases, if 
we remcTaber that the product of two of the factors 
multiplied by itself, is equal to the product of four of 
them ; that the product of four multiplied by the pro- 
duct of two is equal to the product of six ; and that the 
product of four multiplied by tho product of four, is 
equal to the product of eight, &c. Q'hus, in example 2, 
M025 (=l-05xl-05) xi-1025=105xl-05xl-05xl 05. 

EXERCISES. 

1 . What arc the amount and compound interest of 
£91 for 7 years, at 5 per cent, per annura ^ Ans. £12S 
05. l]d. is the amount; and .£37 05. lit/., the com- 
pound interest. 

2. What are the amount and compound interest of 
£142 for 8 years, at 3 per cent, half-yearly.? Ami. 
£227 175. 4ld, is the amount ; and £85 175. 4^^., the 
compound interest. 

3. What are the amount and compound interest of 
£63 55. fi , ars, at 4 per cent, per annum ? Ans. 
£90 05. 5f//.'is the amount; and £26 155. Sfc/., tho 
compound ixiterest. 

4. What are the amount and compound interest of 
£44 05. dd. for 1 1 years, at 6 per cent, per annum .'* 




f\' 



256 



INTEREST. 



Ans. £84 Is. tul. is the amount; aud £39 155. Sd 
the compound interest. "' 

i). What are the amount and compound interest of 
£^2 4s. ^d. for 3 years, at £2 \0s. per cent, halt- 

l ^l^l t''' ^^^ ''• ^■^'^' '^ *^« ^^riouxii- and JC5 
-^s. lUif/., the compound interest. 

r.o^"i ^^'^^^^ ^^^ *^^" amount and compound interest of 
^971 0,; 2\d. for 13 y.^ars, at 4 per cent, per annum } 
Ans in 616 15.S-. 115^^. is the amount; and £645 Ids 
Jid.j the compound interest. 

24. Given the amount, time, aud rate— to find the 
pnncipal ; that is, to find the present icorth of any sum 
to be cire hereafter— a certain rate of interest being 
allowed for the money now paid. 

lluLE.— Find the product of as many factors as there 
are tunes of payment— each _ of the factors bein^ the 
amount of £\ for a single payment ; and divide this 
proau<"t mto the given amount. 

Example.— What sum would produce £834 in 5 years, 
at per cent, compound iateresf? 

TJio amount of XI for 1 year at the given rate is £1-05 ; 

}o-Xl'^^^,^'^''^X^"0^' ^^^"«^ (according to the table) is 

i:831-M-27G28=i:G53 9s. 2,^./., the required principal. 

25. Reason- of thk Rui.k.—Wc have seen [21] that the 
nmoxmt of nny sum is equal to the amount of &\ (for the sanu 
tm.e, and at the same rate) multiplied by tlie principal ; that is, 
the !;L;uroV,£ L ''' ^'^^■^^ pHnoipal==thc given ^incipalX 

r,nl^JI.%J'™v ^'w,'^' ^''''? '^"''^ quani.tics by the same 

number [bee. V. b], the quotients will be equal. Tl.croforc— 

_ Ihe amount of :'.o given principal -f- the amount of £l=thc 

given prmcipalxthe amount of £l-Mhe amount of &\. Tliat 

r' • 1 ] T^T^ ""^ ^^"^ ■?'^'"'' princiijal (the given amount) 
divided by the amount of £1, is equal to the principal, or 
quantity required— which is the rule. i ^^ »^ 



EXERCISES. 

7. What ready money ought to be paid for a debt of 
X629 176-. \\\d., to be duo 3 years hence, alluwin--' 
S i)er cent, compound interest } Ans. j£^500. ° 



INTRRERT. 



257 



8<f., 



H. mat principal, put to interest for G years, would 

0. What sum u'ould produce ^£742 Ids UUl in 14 
years, at 6 per cent, per annum ? Ans. ^32s' 12.. 7d. 

10. ^^ liat IS .€495 19.. ll|r/., to be due in IS years, 
il71 IT'sfd ^^'^^"^'*^^"^'^^' ^^'^^th at present. ^^«.: 

tlio^timo^*'" ^'"^ P''"'cip<'^^ rate, and amount— to find 

^ Rule I.—Divide the amount by the principal: and 
mto the quotient divide the amount of £1 for one my^ 
ment (at the given rate) as often as pcssible-the number 
ol times the amount of £1 U, been used as a divisor, 
will be the required number of payments. 

0\i''il'T'~^'' jvhat time Avill £92 amount to £100 13.. 
V^d., at 3 per cent, half-yearly '? 

XlOG 13.. O'v/.^.:e92=ll.o927. The amount of £1 for 
one payn.ont i. ,£103. Vnit 115927 -- 103 - 11255 • 
1 • 12.),) -^ 103 == l-n'J272 • I0'i'^7'> • 10'^ — i n^^o A 

10009^103^1.03; lS3il^j:;i-\(;eIale^.i t^ 
as a divisor tiinco; therefore the time is 5 pavmenta. or 

S^^vTo-^r'^""'^ *''"'" ^''^^ ^'" ^ ron)ainder after dWid- 
Dip, \>y i Uo, tec. as often as possible 

In explaining the method of tinding the powers and roots 

moni7n?n'^"^''f •^■'- ''i "^'^'^ hereafter, notice a shorter 
method of aseertan.ing ],cnv ofton the amount of one pound 
can Ije used as a divisor. '■ 

27. Rule H.—Divide the given principal by the 
given amount, and ascertain by the interest table in how 
iiumy pnyments £1 would be equal to a quantity nearest 
to the quotient— considered as pounds : this will be tho 
rcquured time. 

KxAMPi.K.—In what time will £50 become £100, at G 
per cent, per atmum compound interest ? 

£100-1-50=2. 

^i^«o^^o''M''^ 1^'? ^f^'^'' *'^''* ^" 1^ years £1 will become 
ro moo '1'-1' ■' ^^'' ' '^'"^ ^" ^- y^^'-' ''"It it ^^'ill l^ecome 
ii '? .; ^; ""^\.'''^ ^^y^^^^ than 2. Tho answer nearest to 
the truth, thereluro. is 12 years. 





li^Si 



^8 



INTKUr.ST, 



28. Rkason ok lliTLK I. — TIio given amount is [20] cquul 
to the givon principal, nuiltiplioil by a proiluct wlucii containa 
as many factors as there arc tijiies of payment— each factor 
being the anioiint of i^l.fur one payment. Hence it is evi- 
dent, that if we divide the given amount by tlie given prin- 
cipal, we must have the product of these factors ; and tiiat, if 
we divide this product, and tlio succesHivo quotients by one 
of the factors, wc shall ascertain their number. 

l^EAsoN OF llui.E 11. — We can find the required number 
of factors (eacli tlie amount uf £1), by ascertaining how often 
the amount of .£1 may bo considered us a factor, withuut 
forming a product tmich greater or less than the quotient 
obtained when we divide the given amount by the given 
principal. Instead, however, of calculating for ourselves, we 
may have recourse to tables constructed by those who have 
already made the necessary multiplications — which saves much 
trouble. 

29. When the quotient [27 J is greater than any 
amount of £\^ at the given rate, in tlie table, divide it 
by the greatest found in the table ; and, if necessary, 
divide the resulting quotient in the same way. Continue 
the process until the quotient obtained is not greater 
than the largest amount in tlie table. Ascertain wliat 
fimiber of jxnjments corresponds to the last quotient, 
and add to it so many times the largest nniuler of pay- 
vients in the table, as the largest amount in the table 
has been used for a divisor 

ExAMPLK. — When would £22 become X535 12s. O^d., 
at 3 per cent, per annum '? 

£535 12s. OJri.-^ 22=24-34500, which is greater than any 
amount of £1, at the ^iven rate, contained in the taljle. 
24-34560-f-4-383l) (the greatest amount of £1, at 3 per cent., 
found in the ta1)lc)=5'55339 ; but this latter, also, is greater 
tluxn any amount of £1 at the giv<-n rate in tlie tables. 
5-55339-i-4'383'J=l'2(iG77, which is found to bo the amount 
of £1, at 3 per cent, per payment, in 8 payments. We 
have divided by the highest amount for £1 in the tables, or 
that corresponding to iifty payments, twice. Therefore, the 
required time, is 50-j-50-f-8 payments, or 108 years. 

EXERCISES. 

11. When would £14 6^. 8^. amount to i218 2s. 8^d. 
at 4 per cent, per annum, compound interest ? Ans. 
In 6 years. 



INTEUKST. 



259 



12. Wlion would jer)4 25. 8^/. amount to £76 35. 5d.^ 
Hi 5 per cent, per annum, compound interest .'* Ans. 
Tn 7 years. 

13. In wliat tinu! would £793 ().?. 2]f/. become J21034 
135. IOJyZ., at 3 per cent, half-yearly, compound interest ? 
Ans. hi 4^ years. 

14. '\Vhcn would £100 become £1639 7.?. 9J., at 6 
per cjLt. halt-yearly, compound interest .'' Ans. In 24 
years. 



QUESTIONS. 

1. What is interest .? [IJ. 

2. AVliat is the diffLjrenco between simple and com- 
pound interest ^ [1]. 

3. AVhat are the principal, rate, and amount ? [1]. 

■ 4, How is the simple interest of any sum, for 1 year, 
found.? [2 &c.]. 

5. How is the simjilo interest of any sura, for several 
years, found } [5]. 

6. How is the interest found, when the rate consists 
of more than one denomination ? [4]. 

7. How is the simple interest of any sum, for years, 
months, &c., found ? [6]. 

8. How is the simple interest of any sum, for any 
time, at 5 or 6, &c. per cent, found .? [7]. 

9. How is tlie simple interest found, when the rate, 
number of years, or both arc expressed by a mixed 
number ? [9J. 

10. How is the simple interest for days, at 5 per cent., 
found .? [10]. 

1 1 . How is the simple interest for days, at any other 
rat«, found ? [H]. 

12. How is the simple interest of any sum, for months 
at f) per cent., found t [12]. 

13. How is the interest of money, left after one or 
more payments, found ? [13]. 

14. How is the principal found, when the amount, 
rate, and time are given .'' [14]. 

1"), How is the" amount found, when the time, ratn, 
and principal arc given ? [15]. 



■hM 



2G0 



DISCOUNT. 



10. ITaw k tliG timo r.uiad, when the amount, prin 
cipal, II ud rate are given ? [10 J. 

17. How is the rate found, when tlio amount, priuci 
pal, and timo arc given ? [18]. 



18. How are the amount, and compound interest found^ 
wlion the principal, rate, and time are given ? [iDj. 

I'J. llow is the present worth of any sum, at com- 
poiuid interest for any time, at any rate, found > [24|. 

20. How is the time found, wlirm t)ie principal, rate 
of compound interest, and amount are given .? [26j. 



DISCOUNT. 

30. Discount is money allowed for a sum paid before 
it is due, and should be such as would be produced hy 
what iii paid, were it put to interest from the time the 
payment is, until the time it ou,o-/ii to he made. 

The presoit loorth of any sum, is that which 
would, at the rate allowed as discount, produce it if 
put to interest until the sum becomes due. 

'M. A bill is not payable until thi-ee days nfter the 
time mentioned in it ; those are called days of grace. 
TIuis, if the time expires on the 11th of the month, the 
bill will no^. be payable until the 14th— except the latter 
falls on a Sunday, in which case it boconies payable on 
the preceding Saturday. A bill at 91 days will not be 
duo until the 04th day after date. 

32. WHicn goods are purchased, ascertain discount is 
oft^Mi allowed for prompt (immediate) payment. 

The discount generally take;, is larger than is sup- 
posed. Thus, lot what is allowed for paying money 
one year before it is duo be 5 per cent. ; in°ordinary 
circumstances ^£95 would bo tho payment for .£100. 
But £\)b would not in one year, at .5 per cent., produce 
more than i299 15.v., which is less than £100 ; the eri-or, 
however, is inconsiderable when the time or sum is small 
Hence to find the discount and present worth at any 
rate, we may ge.nerallu use the following — 



DItCOUNT. 



901 



2?.. Hulk. — Find the interest for the sura to be paid, 
at tlu; dificouut uUowod; consider this ns discount, and 
(hduct it from wha( is due ; the romaindt • will be tlus 
required present woiLh.- 

Example. — £<V"' will ^ ; duo in 3 months ; what should b* 
allowed on inui, lo payment, the discount being at the 
rato of G per cent, per annum 1 

The intorest on £(i2 for 1 year at G per cent, per annum 
is cC3 1 ' 4'l(l. ; and for 3 months it is IHs. l^d. Therefore 
jEi02 miuua iSs. 7|ti.=J(iGl Is. 4^(i., u the required present 
worth. 

.34. To find the present wortli acrMrakly — 

]Ii;le. — Say, as .£100 plus its interest for the given 

tiuio, is to iilOO, ^ ) is the given pum to the required 

present worth. 

TCxAMPLE. — What wnuld, at present, pay a debt nf XI 12 
to he due in (J months, b per cent, per annum disooont being 
allowed ? 

jC 
£ £ s. £ £ i()f) V 142 ^ •'• ^• 

102-5 (100 f-2 10) : 100 : : 142 : — xi^4~=^^^ ^^ ^ 

This is merely a question in a rule already given [14]. 



.1 I 



i 



EXERCISES. 

1. What is the present worth of ^2850 15i-., payable in 
one year, at G per cent, discount > Ans. £802 lis. lO^d 

2. What is the present worth of £240 10.?., payable 
in one year, at 4 per cent, discount ? Ans. £231 5^. 

3. What is the present worth of £550 10s., payable 
in 5 years and 9 months, at 6 per cent, per an. discount > 
Ans. £409 55. loyi. 

4. A debt of £1090 will be due in 1 year and 5 
months, what is its present worth, allowing 6 per cent, 
per an. discount ? Ans. £1004 12s. 2d. 

5. What sum will discharge a debt of £250 175. 6c?., 
to bo due in 8 months, allowing 6 per cent, per an. 
discount .=" A71S. £241 45. 6 J- J. 

6. Wiiat sum will discharge a debt of £840, to be 
duo in 6 montlis, allowing 6 per cent, per an. discount ? 
A71S. £815 IO5. HUl 




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15ISC0UNT. 



7 What ready money now will pay a debt of £200, 
to bo due 127 days hence, discounting at 6 per oent 
per an.? Am. £ldo iSs. 2-}d. f « ut. 

8.^ \Vhat ready money now will pay for ^1000, to be 

9 A bill of £150 105. will become due in 70 days 
what ready money will now pay it, allowing 5 per cent 
per an. discount ? Ans. dei49 Is. bd 

10. A bill of £140 10,. will be due in 76 days, what 
ready money will now pay it, allowing 5 per cent, per 
an. discount > Ans. £139 1,. O^'Z. ^ 

11. A bill of £300 will be due in 91 days, what wiU 
^oTu?^^ ^\ al owing 5 per cent, per an. discount .? Ans. 

jb29D <ys. l^d. 

12. A bill or £39 5^. will become due on the first 
ot beptember, what ready money wUl pay it on the 

^^£38 iS 1-5"^^' ^^^''''''° ^ ^'' "'''*• P"" ^"- • 

13. A bill of £218 3.9. SicZ. is drawn of the 14th 
August at 4 months, and discounted on the 3rd of Oct • 
what IS then its worth, allowing 4 per cent, per an! 
discount .? Ans. £216 Ss. \id. 

14 A bill of £486 185. 8^. is drawn of the 25th 
March at 10 months, and discounted on the 19th June 
what then is its worth, allowing 5 per cent, per an' 
discount.? Ans. £412 9s. U^d. e ■ 

15. What is the present worth of £700, to be due in 
9 months, discount being 5 per cent, per an. > Ans 
£674 135. 11^^. ^ p u. . Jins. 

16. What is the present worth of £315 12, 41^/ 
payable in 4 years, at 6 per cent, per an. discount"? 
Ans. £254 lO.e. 7]-d. 

17 What is the present worth and discount of £550 
105. for 9 months, at 5 per cent, per an. } Ans. £530 
125. Q\d. is the present worth; and £19 175. lli^; 
s the discount. * * 

18. Bought goods to the value of £35 135. 8^. to be 
Daul m 294 days; what ready money are they now 
ivorth, 6 per cent, per an. discount being allowed ? 
Ans. £31 05. 9^^/. ^ 



COMMISSION. 



263 



19. If a legacy of £600 is left to me on the 3rd of 
May, to be paid on Christmas day following, what must 
I receive as present payment, allowing 5 per cent, per 
an. discount.^ Ans. i:i581 4s. 2}d. 

20. What is the discount of £756, the one half pay- 
able in 6, and the remainder in 1.2 months, 7 per cent, 
per an. being allowed ? Ans. £37 I4s. 2\d. 

21. A merchant owes £110, payable in 20 months, 
and £224, payable in 24 months ; the first he pays in 5 
mouths, and the second in one month after that. What 
did 1)0 pay, allowing S per cent, per an. } Ans. £300. 

QUESTIONS FOR THE PUPIL. 

1. What is discount .? [30]. 

2. Wliat is the present icorth of any sum } [30]. 

3. \s'\\iii QXQ. days of grace] [.?1]. 

4. How is discount ordinarily calculated } [33] 

5. How is it accMrately calculated > [34] . 



COMMISSION, &c 



3.5. Commission is an allowance per cent, made to a 
person called an agent., who is employed to sell goods. 

Insurance is so mucli per cent, paid to a person who 
undertakes that if certain goods arc injured or destroyed, 
he will give a stated sum of money to the owner. 

Brokerage is a small allowance, made to a kind of 
agent called a broker, for assisting in the disposal of 
goods, negotiating bills, &c. 

36. To compute commission, &c. — 

Rule. — Say, as £100 is to the rate of commission, so 
Is the given sum to the corresponding commission. 

ffc- Example. — What will be the commission on goods worth 
£437 56-. 2</., at 4 per cent. ] 

£100 : £4 : : £437 5s. 2d. : l^l^iJIii-?^ = £17 9^. 



100 



9i'<Z., the required couimissiion. 

V7.V 37. To find what insurance must be paid so that, if 
the goods are lost, both their value and the insurao'io 
paid mr.y be recovered — 



264 



COMMISSIOI* 



Rule.— Say, as £100 minus the rate per cent, is to 
eClOO, so is the value of the goods insured, to the 
required insurance. 

Example.— What sum must I insure that if goods worth 
i.4UU are lost, I may receive both their value and the in- 
surance paid, the hitter being at the rate of 5 per cent '* 

£95 : £100 :: £400 : ^122^0=^421 1. 0^^ 

If £100 were insured, only £95 would be actually received, 
since £5 was paid for the £100. In the example, £421 Is Ohd 
are received; but deducting £21 Is. OU, the insurance, £400 
remains, 

EXERCISES. 

1. What premium must be paid for insuring goods 
to the amount of £900 15s., at 2^ per cent, f A7is 
£2,2 105. 4ir/. ' - f 

2. What premium must be paid for insuring goods 
to the amount of £7000, at 5 per cent. ? Ans. £350 

3. What is the brokerage on £976 175. 6d., at 55. 
per cent. > Ans. £2 Ss.'lQid. 

4. What is the premium of insurance on goods worth 
£2000,^ at H per cent. ? Ans. £150. 

5. ^Vlmt is the commission on £767 145. 7d , at 2i 
pcrcent. .? A7is. £19 3s. lO^d. ^ 

y,^',^^'''^ ^'^'^^ ^^ *^^e commission on goods worth 
i'J71 145. 7rf., at 5.5. per cent. ? Ans. £2 8s. 7-^-d 

7. What is the brokerage on £3000, at 25. 6^. per 
cent. ? Ans. £3 15s. ^ 

S How much is to be insured at 5 per cent, on goods 
worth £900, so that, in case of loss, not only the value 
ot the goods, but the premium of insurance also, may bo 
repaid ? ^ Ans. £947 75. 4/-^. ' -^ 

9 Shipped off for Trinidad goods worth £2000, how 
much must be insured on them at 10 per cent., that in 
case of loss the premium of insurance, as well as their 
value, may be recovered ? Ans. £2222 45. dhd. 



QUESTIONS FOR THE PUPIL. 



1 



What is commission ? [35], 

2. What is insurance ? [35]. 

3. What ia brokerage ? [35] 






PURCHASE OF STOCK. 



265 



0?rf. 



at24- 



4. IIow are commission, insurance, &c., calculated? 

[36]. 

5. How is msnrancG calculated, so that both the in- 
surance and value of the goods may be received, if tho 
latter are lost ? [37] . 



PURCHASE OF STOCK. 



.>o. 



Stock is money borrowed by Government from 
individuals, or contributed by merchants, &c.,^ for the 
purpose of trade, and bearing interest at a fixed, or 
variable rate. It is transferable either entirely, or in 
part, according to the pleasure of the owner. 

If the price per cent, is more th;in £100, tlie stock in 
question is said to be ahave^ if less than i^lOO, helow " par." 

Sometimes the shra-es of trading companies are only 
gradually paid up ; and in many cases the whole price 
of the sliare is not demanded at all — they may be ^£50, 
£100, &c., shares, while only £5, £10, &c., u.ay have 
been paid on each. One person may have many shares 
When the intesest per cent, on i\\Q money paid is con- 
.sidera1)lo, stock often sells for more than what it origi- 
nally cost; on the other hanu, when money becomes 
more valuable, or the trade for which the stock was 
contributed is not prosperous, it sells for less. 

39. To find the value of any amount of stock, at any 
rate per cent. — 

EuLE. — Multiply the amount by the value per cent., 
and divide the product by 100. 

ExAMPi.K.— When £G'J \ will purchase £100 of stock, what 
will purchase £G42 ? 



£G42x69j 
100 



-=£443 15s. lid. 



It is evident that £100 of stock is to any other amount of 
it, as the price of tho former is to that of tho hxttor. Tims 

£100 : £612 :: £69 .\ : ^il^^.^ 

100 

EXERCISES. 

1. What must be given for £750 16i\ in the 3 per 
cent, annuities, when £64 j- will purchase £100 .? Ans. 
£481 95. O^^V^ 



■I 



iii*( 



2Rf) 



EQUATION OP PAYMENTS. 



2. What must be given for ^£1756 Is. 6d. India stock, 
when £]U6l will purchase dt^lOO ? Ans. £3446 17s. i<^d 

3. What is the purchase of ^29757 bank stock,' ai 
J212oA per cent. ? Ans. jei2257 4^. 7^d. 

QUESTIONS. 

1. What is stock .? [38]. 

2. When is it above, and when below " par" ? [38], 

3. How is the value of any amount of stock, at an^ 
rate per cent., found ? [39]. 



EQUATION OF PAYMENTS. 

40. This is a process by which we discover a time, 
when several debts to be due at dij'ereni periods maybe 
paid, (il once, without loss either to debtor or creditor 

lluu.:.--Multiply each payment by the time which 
should elapse before it would beco-ie due ; then add 
the products together, and divide their sum by the' sum 
01 the debts. 

Example 1. -A person owes another £20, payable in 6 
months; i 50 payable in 8 months; and X90 payable in 
U months. At what time may all be paid together, without 
OSS or gam to either party '? * o ) 

il jl 

20 X 0= 120 
/30x 8= 400 
_90x 12=1 080 

IGU 1GO)TOOO(10 the required number of mor'^g. 
160 

. ExAMPLK 2.— A debt of £450 is to be paid thus : £100 
immediately, £300 in four, and the rest in six months \V lien 
Bhould it be paid altogether ? 



£ 

100 
300 
_50 

450 



£ 
X 0= 

X 4=1200 
X 6=^03^ 
450)1500(31 months 
1350 

"iso 

450 



EQUATION OP PAYMKNTS. 



•267 



41. Wo liavG (according to a i^rinciplo fonnorly used 
[13]) reduced each debt to a sum which would bring the 
same interest, in one month. For G times i^20, to be due 
in 1 month, should evidently produce the fsame as £20, to 
be due in G months — and so of the other debts. And the 
interest of j£lGOO for the smaller time, will just be equal to 
the interest of the smaller sum for the larger time. 

EXERCISES. 

1. A owes B jeeOO, of which £200 is payable in 3 
months, £150 in 4 months, and the rest in 6 months ; 
but it is agreed that the whole sum shall be paid at 
once. When should the payment be made ? Ans. In 
41 months. 

2. A debt is to be discharged in the following man- 
ner : I at prcijent, and ^ every three months after antil 
all is paid. What is the equated time ? Ans. A\ 
months. 

3. A debt of £120 will be due as follows : £50 in 
2 months, £40 in 5, and the rest in 7 months. "When 
may the whole be paid together 't Ans. In 4^ months. 

4. A owes B £110, of which £50 is to be paid at 
the end of 2 years, £40 at the end of 3^, and £20 at 
the end of 4-1^ years. When should B receive all at 



once .'' Ans. In 3 years. 

5. A debt is to be discharged by paying ^ in 3 months, 
i- in 5 months, and the rest in 6 mouths. What is the 
equated time for the whole .'' Ans. 4| months. 

QUESTIONS. 

1. "What is meant by the equation of payments } 

2. What is the rule for discovering when money, to 
be due at different times, may be paid at once } [40] . 



! ■ 



' ii 
i. 



208 



SECTION VIII. 



EXCHANGE, &c. 

1. Exchange enables us to find what amoimt of the 
inoncy of one country is equal to a given amount of the 
money of another. 

Money is of two kinds, real— or coin, and imaginary— 
or money of exchange, for which there is no coin ; as, 
lor example " one pou7id sterling." 

The par of exchange is that amount of the money 
of one country aduallt/ equal to a given sum of tho 
money of another ; taking into account the value of 
the metals they contain. Tho mirse of cxchan<rc if^ 
that sum which, in point of fact, would be allowed 
for it. 

2. When the course of exchange with any plac6 is 
a^ove ;' par," the balance of trade is against that place. 
Thus if Hamburgh receives merchandise from London 
to the amount of ^£100,000, and ships off, in return, goods 
to the amount of but c£50,000, it can pay only half what 
It owes by bills of exchange, and for the remainder must 
obtain bills of exchange from some place else, giving 
for them a premium— which is so much lo?.t. IJut the 
exchange cannot be much above par, since, if the pre- 
mium to bo paid for bills of exchange is high, tho 
merchant will export goods .it loss profit ; or Tie will 
pay the expense of transmitting aixl iusuriu*' coin, or 
bullion. ° 

3. The nominal value of commodities in these countries 
was from four^ to fourteen times less formerly than at 
present ; that is, the same aiJ^ount of money would then 
buy much more than now. We may estimate the value 
of money, at any particular period, from the amount of 
corn It would purchase at that time. The value of 
money fluctuates from the uature of the crops, the statu 
of trade, &o. 



KXCHANGE. 



209 



111 ; a.s, 



la cxcliango, a variublo is given for a fixed sum ; ihiin 
LonJf)!! receives difioreut values for £1 from diliereut 
countries. 

Agio is the dilTerence wbieli there is in some places 
between the cwrreiU or msk money, and the uxkange 
or hank money — which is finer. 

The following tables of foreign coins arc to be mad'.' 
familiar to the pupil. 



FOREIGN MONEY. 

MONEY OF AMSTERDAM. 

Flemish Money. 

• . make 1 groto or penny. 

• . • 1 stiver. 
• 1 florin or guilder 



Penningii 



16 or 

320 

800 
1920 



giote« 



40 or 



100 
240 



stivers 
20 



50 or 
120 or 



guilders 

2i 
6 



1 rixdollar. 
1 pound. 



rfenningg 
6 



72 or 
M40 



g rotes 
12 



MONEY OF HAMBURGH. 

Flemish Money. 
I • . make 1 grote or penny 



lings 



1 skilliug. 
1 pound. 



Ffenmnga Tenco 
12 or 2 



192 

884 
676 



I skilli 
240 orf 20 

Ilamhv,rgh Money. 

make 1 scliilling, equal to 1 stiver 
1 mark. 



32 or 

64 
96 



schillings 
16 



32 or 
48 or 



marks 
2 
3 



1 dollar of exchange. 
1 rixdollar. 



We find that 6 scliillings=l skilling 

Hamburgh money is distinguished by the word " Harabro." 
" Lub," from Lubec, where it was coined, was formerly used 
for tliis purpose ; thus, '• one mark Lub." 

Wo exchange with Holland and Flanders by the pound 
Bt^cling. 

N 



J-iU 



rM 



i>70 



KXCIIAWaE. 



KRKJSCn MONKY. 



Dernioi!* 
12 



Accouula wcro Ibnuerly kept in livrus, &c. 



210 or 
720 



Centimes 
10 



sous 

20 



make 1 sou. 



1 livre. 



livres 
60 or I 8 



1 ecu or crown 
AcoountH are now kept in francs and centimes. 

• . make 1 dccime. 



dccimei 
100 or I 10 . 
81 livrea=80 franca. 



1 franc. 



llocs 
400 

1000 or 
4800 



PORTUGUESE MONEY. 

Accounts are kept in milrees and recs. 

«... make 1 crusado. 

crnsadoi 
2i . 
12 ... . 



1 milree. 
1 moidore. 



SPANISH MONEY. 



Spanish money is'of two kinds, plate and vellon ; the latter 
being to the former as 32 is to 17. Plate ia used in exchange 
with us. Accounts are kept in piastres, and maravedi. 



Maravedies 
84 



make 1 real. 



272 or 

1088 
375 



reals 
8 



piastres 
32 or I 4 . 



1 piastre or piece of eight 

1 pistole of exchange. 
. 1 ducat. 






AMERICAN MONEY. 

In some parts of the United States accounts are kept in 
dollars, dimes, and cents. 
Cents 
10 . . , , , make 1 dima 

idimea 
10 . . . . . 1 dollar. 

In other parts accountg are kept in pounds, shillings, and 
pence. Those are called currency, but they ar« of much less 
yalue than with ua, paper money being usad. 






Pf<«nninjj!i 
12 



KXCHANtiS 



DANISH MONET. 



i?71 



make 1 skill! u;;. 



102 or 



nkillin 
10 



K» 



marki 
1152 OC or (} . 

amburgU marks. 

VKNETIAN MONEY. 
Dnnari (the plurnl ordqnaro) 

12 .... innke 1 soldo, 

soldi 

210 or JiO . 

liro golili 

1188 121 or "6 4 

l'J20 100 8 . . 



1 mark. 
1 rixdollar 



1 lira. 

1 ducftt current. 
1 du(^t ellcctivo 



AUSTRIAN MONEY. 



rfcmiiiigs 
4 



210 or 

800 



Grains 
10 



oroutzers 
00 __. 

fioring 



90 or I U 



NEAPOLITAN MONEY. 



cailius 

100 or I 10 



mako 1 croutzsr 
1 florin. 
1 rixdollar. 

make 1 carlin. 

1 ducat rtt,A9 



MONEY OF GENOA. 
Lire soldi 

4 nnd 12 make 1 scudo di cambio, or crown of exchange. 
10 nnd 14 1 scudo d'oro, or gol I crown. 



Dcnari di pe/.za 
12 



OF GENOA AND LEGHORN. 

mako 1 soldo di pozza. 



I soldi di pez/a 
20 



f)cnarj di lira 
12 



240 or 
1380 



soldi di lira 
20 . 
110 or 1 5| 

SWEDISH MONEY. 



1 pezza of 8 reals, 
make 1 soldo di lira. 



Fonnings, or oers 
12 



Iskillingi 
48 



1 lira. 

1 pezza of 8 reals 

make 1 skilling. 
1 rixdoUai 



I 



f ■ 




ft! 



ill 



ii i 



I 



t73 



CXOIANOE. 



RUSSIAN MONEY. 



160 



mnko 1 ruble. 



EAST 



niuko 1 rupco. 



DIAN W0NE1 
Towriei 

Kiinoci 

100,000 . . . iiao. 

10,000,000 . . 1 croro. 

The cowrie is a small ehoU found at the MaMivuH, and near 
Anj;ola : iu Africa about 5000 of them pass for a pound. 

The rupcq lias different values : at (Juloutta it is 1;». 11 j,/. 
tbo Sicca rupee is 2s. OU. ; and the current rupee 2.*.— if wo 
divide any number of tlicse by 10, we change them to pounds 
of our money; the Boinbny rupee is 2s. {5^/., &c. A sum of 
Indian money is expressed as follows; 5-88220, which means 
5 la(!3 aiid«8220 rupees. 



d. To rcduco bank to current money — 

Ki'i.K.— Say, as J2100 is to JL'IOO + the agio, so is 
the given amount of bank to tlio recjuired amount of 
current money. 

KxAMi'LK. — How many c;uildor.«<, current munoy, arc equal 
to 403 ouildors, 3 stivers, and 13!jt ponnings banco, a^io 
being 4^' < .0 a 

1<J'^ : lO^ : : 403 g. 3 st. 13«4 p. : 1 ' 

7 7 20 ^"^ 



TOO 
05 



733 



1)203 stivers. 
10 



45500 



14S221 pcmiinga. 
Multiplying by 05, and adding 04 to tho 
will give 0034429 I'l'^^li'^t, 
]Multi])lying by 733 
and dividing by 45500)T0020lio4r)7 

will give 155200 penniugs. 
10)155209 

20 )9700 9 
And 485 g. O"^ d^tH p. is the amount sought. 
5. We multiply the first and second terms by 7, and add tha 
numerator of the fraction to one of tlie products. This is tlie 
same thing as reducing these terms to fractions liaving 7 for 
their denominator, and then multiplying them by 7 [Sec. V. 29] 
For the same reason, and in tho same way, we multiply the 
first and third terms b/ 65, to banish tho fraction, without 
aeitroying the proportion. 



eXCHANOE. 



273 



TIic remainder of tho process i« nccording to Iho rulo of 
lnoporUofi [Soo. V. 1)1]. VVc roduco tho nnawcr to pcnaingH, 
BtiverH, mill jjniMoia. • 

EXKHCIHF.S. 

1. R<m1iico n7-l ^•uildofH, 12 Htivors, banlf monoy, to 
cm rent money, agio being 4i per cent. ? Am. 31)2 g., 
5 St., :},V, p. 

2. llecluce 4378 guililers, «< stivers, bank money, to 
current money, agio being 4* per cent. ? Avs. 4577 «., 
17 St., r^Vs p. 

3. Ueduee S73 guilders. 1 1 stivers, bank money, to 
current money, agio being 4 J per cent. ? A7is. UIG g., 
2st., HJap. 

4. llediicc 1012 guilders, bnnk monoy, to current 
money, agio being 4|i per cent. ? Am. 1722 g., 14Ht., 
lOA p. 

6. To reduee current to Itank money — 
_ liiu.i:. — Say, as JUlOO-f-thc agio is to JCIOO, so Is tlio 
given amount of current to the required amount of 
baidc money. 

ExAMi'i.K. — How much bank money is thcro in 485 guil- 
ders and ^J'ii'iol pennings. agio being 4^' i 



104? 
7 

733 

4550c 



100 

7 

700 



g. St. p. 

20 



33351500 



yioo 

10 
15520!) 



Multiplying by 45500 tho denominator, 

7002009500 
and adding 25957 tho numerator, 

we get 7002035457 

700 

33351500)4943424819900 
Qu(7tIenl~lT822 1 fl 
10) 1 48221;; I 
20)9203 
403 3 13^^ is tho amount soug*i! 



i 




I 



274 



EXCHANttE. 



EXERCISES. 

5- Reduce 58734 gi^lders, 9 stivers, 11 penningR, 
current money, to Lank money, agio being 4^ per cent. ? 
Ans. 560P6 g., 10 St., llJfi p. 

6. lieduce 4326 guilders, 15 pcnnings, current money, 
to bank money, agio being 4-f per cent. ? Ans. 4125 g., 
13st.,2i||p. 

7. Eeduce 1186 guilders, 4 stivers, 8 pennings, cur- 
rent, to bank money, agio being 4f per cent. } Ans 
1136 g., 10st.,0iff p. 

8. Keduce 8560 guilders, 8 stivers, 10 pcnnings, 
current, to bank money, agio being 4i per cent. . 
Ans. 8183 g., 19 st., 5fi3. p. 

7. To reduce foreign money to Ikitish, &c. — 
BuLE. — Put the amount of British money considered 
in the rate of exchange as third term of the proportion, 
i^^' value in foreign money as first, and the foreign 
money to be reduced as second term. 

Example 1. — Flemish Money. — How much British money 
is equal to 1054 guilders, 7 3tiy~-s, the excliance bcine; 33*-. 
4d. Hemish to £1 British 1 

S3.S. 4.,;. : r054 g. 7 st. : : £1 : ? 
12 20 



4U0 pence. 



21087 stivers. 
2 

400)42174 Flemish pence. 
_£10r,435 = £105 8s. 8iJ. 
£1, the amount of British money considei'cd in the rate, 
is put in the third term , 335. 4d.. its value in foreign money, 
in the first; and 1054 g. 7 gt . the money to be reduced, 
in the second. 

9. How many pounds sterling in 1680 guilders, at 
335. 3d. Flemish per pound sterling ,? A71S. JE168 8s. 

10. Reduce 6048 guilders, to Rritisli money, at 33.?. 
\\d. "Flemish per poun'i British .? Ans. i.'594 7a-. 

-It T-» 1 

XI. jL'teuuco 



money, at 34^. 
£198 85. 61 f^^, 



W.7 
'Jit, 



04S guilders, L. sUveis, to British 
Flemish per pound sterling > Am 



M'r 



EXCHANGE. 



375 



lit. npw many pounds sterling in 1000 guilders, 10 
stivers, exchange being at 335. 4d. per pbuud sterling ? 
Jbis. iiilOO Is. * 

Example 2. — Hamburgh Money. — How much British 
money is equivalent to 476 marks, 9 skillings, the exchange 
being 33s. iid. Flemish per pound British '? 

s, d. m. 8. 

33 6 : 476 9| : : £1 : I 

12 32 2 



ling > 

16. Reduce 



402 grotes. 15232-f 19'=15251i grotes. 
402 )152511 

£37-9386=£37 I85. 9d. 

Multiplying the schillings by 2, and the marks by 32, 
reduaes both to pence. 

13. How much British money is equivalent,to 3083 
marks, 12| schillings Ilanibro', at 325. 4d. Flemish per 
pound sterling .? Ans. £254 65. 8^^. 

14. How much English money is equal to 5127 marks, 
5 Schillings, Hambro' exchange, at 36s. 2d. Flemish 
per pound sterling ? uins. i£378 Is. 

15. How many pounds sterling in 244S marks, 9|- 
schillings, Hambro', at 32^. 6d. Flemish per pound ster- 

A/is. £200 105. 

7854 marks, 7 schillings Hambro*, to 
British money, exchange at 345. lid. Flemish per 
pound sterling, and agio at 21 per cent. ? Ans. £495 
Ids. Old 

Example 3. — French Money. — Reduce 8654 francs, 42 
centimes, to British money, the exchange being 23f., 50c., 
per £1 British. 

f. c. f. 0. 8654-42 

23 50 : 8054 42 : : 1 : ■-23:50=-=^368 5s. 5^. 

42 centimes are 042 of a franc, since 100 centimes make 
1 franc. 

17. Reduce 17969 francs, 85 centimes, to British 
money, at 23 franc \, 49 centimes per pound sterling > 
Ans. £765. 

18. Reduce 7672 francs, 50 centimes, to British 
money, at 23 francs, 25 centimes per pound sterling ? 
Ans. £330. 




276 



EXCHANGE. 



10. EetlucG 1,5647 francs, 36 centimes, to British 
money, at 23 francs, 15 centimes per pound sWlinf/ ? 
A71.S. £675 ISs. 2ld. • ^ ^ o 

20. lieduce 450 francs, 58^^ centimes, to British 
money, at 25 francs, 5 centimes per pound sterling > 
Ans. £nQ Us. ^ ' 



Example 4. 



■ Pmiuguese 



Money. —RovT much British 
money is equa to 540 milrees, 420 rees, exchange beino; at 
OS. m. per milree ? o o 



m. m. r. s. d. 

1 : 540-420 : : 5 G : 540-420x5s. 6c^.=£148 125. 3,^^. 

^,V!".^^^^® *^^*^ S>^i*i»h money is the variable quantity, 
and OS. M. is that amount of it which is considered in the 

The rees are clianged into the decimal of a milree bv 
putting them to the right hand side of the decimal point 
since one reo is the thousandth of a milree. 

21. In 850 milrees, 500 rees, how much British 
money, at 55. 4d. per milree } Ans. £22Q \Qs. 

22. Reduce 2060 milrees, 380 roes, to English money 
at 55. Q^d. per milree } Ans. ^£573 0^. lOi^. 

23. In 1785 milrees, 581 rees, how m*any pounds 
sterling, exchange at 64i per milree.? Ans £479 
175. Qd. 

24. In 2000 milrees, at 5^. 81^^. per milree, how 
many pounds sterling.? Ans. £570 165. S^^. 

Example 5 —5;,_aH?5/i ilfonei/.— Reduce 84 piastres, 6 reals, 
IJ maravedi, to British money, the exchange beino- 4Ur/ the 
piastre. ® o • ^"" 



r. 

6 



8 



m. 
19 



d. 

40 



8 
34 

272 



678 reals. 
34 

23052 maravedi. 
49 



272)1129548 



4152-7, &c.=£17 %:. 02d. 



EXCHANGE. 



277 



EXERCISES. 

25. ReducG 2448 piastres to British money, exchange 
at 50^. sterling per piastre ? Ans. i£;510. 

26. Keduco 30000 piastres to British money, at 40d. 
per piastre ? Ans. £5000. 

27. Reduce 1025 piastres, 6 reals, 22if ^ maravedi, to 
British money, at :^9ld. per piastre ? Ans. i2167 15s. 4d. 

Example 6. — American Money. — Reduce 37G5 dollars to 
British money, at 4s. per dollar. 4s.=£\ : therefore 
5 )37G5 dol. del. s. £ 

753 is the required sum. Or 1 : 3765 : : 4 : 753 

28. Reduce £292 3.?. 2^d. American, to British money, 
at 66 per cent. .? Am. £176. 

29. Reduce 5611 dollars, 42 cents., to British money, 
at 4s. d^d. per dolLir ? Ans. £1250 175. 7d. 

30. Reduce 2746 dollars, 30 cents., to British money, 
at 45. S^d. per dollar ? Aqis. £589 Gs. 2^d. 

From these examplcM the pupil will very easily under- 
stand how any other hind of foreign, may be changed 
to British money. 

8. To reduce Britisli to foreign money — 
Rule. — Put that amount of foreign money which is 
considered in the -rati* of exchange as the thjrd term, 
its value in British money as the first, and the British 
money to be reduced as the second term. 

ExAMPLK 1. — Flemk'i Money. — How many guilders, &o., 
in jC2oG 149. 2:1. Britisli, the exchange being 34s. 2d. Flemish 
to £1 British I 

£ £ s. d. s. d. 

1 : 23G U 2 :: 34 2 : ? 
20 20 12 



20 
12 

240 



4734 
12 

5G8UM. 
410 



410 pence. 



24Qyr>202100 

T2" )97050-4, &c. 
20)8087 _ji_ 

iC404 7 U Flenieh. 



N 2 



278 



EXCHANGE. 



We might take parts for the 34.s. 2d.— 

345. 2d.=£l 4- 10s.4-4s.+2c/. 

£> £ s. d. 

^ei = 1 23G 14 2 

I0s.= i 118 7 1 

47 6 10 
5i 



i 

4s.=a ^ 

2^^"=rL (aV of 1) 1 19 



£404 7 61 Flemish. 

EXERCISES. 

31. In £100 l5., how much Flemish money, exchan<ro 
at 335. 4d. per pound sterling? Am. 1000 guilders, 
10 stivers. ' 

32. Reduce £168 85. dji^d. British into Flemish, 
exchange being 335. 3d. Flemish per pound sterling ? 
uItw. 1680 guilders. ^ " 

33. In £199 ll5. 10^/j^.Britisli, how much Flemish 
J!!?i!?^' exchange 345. 9^. per pound sterling ? Ans. 
2080 guilders, 15 stivers. 

34. Reduce £198 85. e^d. British to Flemish 
inoney, exchange being 345. 5d. Flemish per pound 
sterling > Ans. 2048 guilders, 15 stivers. 

in £'9aT''p^-7?""*T^^ JMovj^i/.-How many marks, &c., 
in £24 65. British, exchange being 335. 2d. per £1 British l 



£1 

20 

20 



£24 65. 
20 

486 
398 



33s. 
12 



2d. 



398 grotes. 



20 )193428 
2)9671 8 pence. 
16 )4835 schillings, 1 penny. 

302 marks, 3 schillings, 1 penny. 
35. Reduce £254 65. 8d. English to Hamburgh 
money, at 325. 4d. per pound sterling.? ]Ans. 3083 
marks, 12| stivers. 

36 Reduce £378 I5. to Hamburg money, at 365 
2d. Flemish per pound sterling ? Ans. 5127 marks. 
5 schillings. _ ' 

37. Keduce £536 to Hamburgh money, at 865. 4d 
per pound sterling .? Ans. 7303 marks. 



EXCHANGE. 



279 



38. Reduce JB495 155, OJ<Z. to Hamburg currency, 
at 345. lid. per pound sterling ; agio at 21 per cent. ? 
Ans. 7854 marks 7 schillings. 

Example S. — French Moncij. — How much French money 
is equal in value to £83 2s. 2d., exchange being 23 francs 
25 centimes per £1 British ^ 

£ £ s. d. t 

1 : 83 2 2 : : 2325 : ^ 

20 20 



20 1662 
12 12 



240 19946 
23-25 

240 )463744-50 

19322-7, or 19322f. 70c. is the required sum 

39. Reduce £274 55. Oc?. British to francs, &c., ex- 
change at 23 francs 57 centimes per pound sterling r 
Alls. 6464 francs 96 centimes. 

40. In £765, how many francs, &c., at 23 franca 
49 centimes per pound sterling } Ans. 17969 franca 
85 centimes. 

41. Reduce £330 to francs, &c., at 23 francs 25 cen- 
times per pound sterlijjg .? Ans. 7672 francs 50 cents. 

42. Reduce £734 45. to French money, at 24 franca 
1 centime per pound sterling } Ans. 1769 francs 42-J- 
centimes. 

Example 4. — Portuguese Money. — How many milrees anrf, 
rees in £32 6s, British, exchange being 5s. 9(/. British pe 
milree 1 

s. d. £ s. 
5 9 : 32 G : : 1000 : ? 
12 20 



69 



646 
12 

7752 
1000 



69)7752000 
,«quircd sura 



112348 rce8=112 milrees 348 rees, is tno 



260 



XXCHANGE. 



43. Reduce £226 16^. to milrecs, &c., at 5^. 4d. per 
milrce ? Ans. 850 milrees 500 roes. 

44. Reduce £^479 17*. 6d. to milrees, &c., at 6Ud 
por milrec ? Ans. 1785 milrees 581 rees. ' * 

45. Reduce £570 16.9. 8^. to milrees, &c., at 5*. Sid 
per milree ? Ans. 2000 milrees. 

46. Reduce £715 to milrees, &c., at 5*. 8d. permU- 
ree ? Ans. 2523 milrees 529/^ rees. 

. ^^£^S^F,.5--'Si'«ww;i Morm/.— Row many piastres, &o., 
in £02 British, exchange being 50d. per piastre '» 

d. £ 

50 : 62 : : 1 : ? 
20 



1240 p. r. m. 
12 ^97 32if , is the required sum. 



50 )14880 

^ 297-6 piastres. 
8 

48 reals. 
34 



50)1632 



32M maravedis. 



•*7. How many piastres, &c., shall I receive for £510 
sterling, exchange at 50c?. sterling per piastre ? Ans. 
2448 piastres. 

48. Reduce £5000 to piastres, at 40^. per piastre > 
Ans. 30000 piastres. 

49. Reduce £167 15*. 4d. to piastres, &c., at SO^d. 
per piastre ? Ans. 1025 piastres, 6 reals, 22^4^ mara- 
vedis. ' 

^ 50,. Reduce £809 95 8d. to piastres, &c., at 40|J. per 
piastre ? Arts. 4767 piastres, 4 reals, 2yVV maravedis. 

Example Q.— American Money .—Reduoe £176 British to 
American currency, at 66 per cent. 

£ £ £ 

100 : 176 :: 166 ; : 

166 
100)29216 



£292 35. 2i(/., is the required sum. 



EXCHANGE. 



281 



EXERCISES. 

61. Reduce £753 to dollars, at 4s. per dollar > Ans. 
3765 dollars. 

52. Ileduce ^£532 4s. Sd. British to American money, 
at 64 per cent. > Ans. £872 175. 3d. 

53. Ikduce £1250 17s. 7d. sterling to dollars, at 
4$. 5^d. per dollar ? Ans. 5611 dollars 42 cents. 

54. Ileduce £589 6s. 2^%d. to dollars, at 4s. S^d. 
per dollar } Ans. 2746 dollars 30 cents. 

65. Reduce £437 British to American money, at 78 
per cent. ? Ans. £777 17s. 2^d. 

9. To reduce florins, &c., to pounds, &c., Flemish — 
Rule. — Divide the florins by 6 for pounds, and — 

adding the remainder (reduced to stivers) to the stivers 
—divide the sum by 6, for skillings, and double the 
remainder, for grotes. 

Example.— How many pounds, skillings, and grotes, in 
105 florins 19 stivers '? 

f. St. 
6)165 19 
£21 13s. 2d., the required sura. 
6 will go into 1G5, 27 times— leaving 3 florins, or 60 stivers, 
"which, with 19, make 79 stivers ; 6 will go into 79, 13 times- 
leaving 1 5 twice 1 are 2. 

10. Reason of ^he Rule.— There are 6 times as many 
florins as pounds ; for we find by the table that 240 grotea 
make £1, and that 40 C^*") grotes make I florin. There are 
6 times as many stivers as skillings ; since 96 penniugs^make 

1 skilling, and 16 (V) pfennings make one stiver. Also, sinca 

2 grotes make one stiver, the remaining stivers are equal to 
twice iiH many grotes. 

Multiplying by 20 and 2 would reduce the florma to grotes ; 
and dividing the grotes by 12 and 20 would reduce thorn to 
pounds. Thus, using the same example— 
f. St. 
165 19 
20 



3319 
2 

12 )6638 - 
20)653_ 2 

£27 ]3s. 2d., as before, is the result. 



f ' 



?l 



« 



282 



EXCHANGE. 



EXERCISES. 

56. Ill 142 florins 17 stivers, how many pounds, &c., 
Atis. £23 16*. 2d. 

57. lu 72 florins 14 stivers, how many pounds, &c., 
Ans. £\2 2s. 4(1. 

58. In 180 florins, how many pounds, &c. } Am. iE30 
11. To reduce pounds, &o., to florins, &c. — 
Rule. — Multiply the stivers by 6 ; add to the producfi 

half the number of grotes, then for every 20 contained 
in the sum carry 1, and set down what remains above 
the twenties as stivers. Multiply the pounds by 6, and, 
adding to the product what is to be carried from the 
stivers, consider the sum as florins. 

Example. — How many florins and stivers in 27 pounds, 
13 skillinga, and 2 grotes ? 

£f s. d. 
' 27 13 2 . 

6 



165fl. 198t., the required sum. 
6 times 13 are 78, which, with half the number (f ) of 
grotes, make "^0 stivers — or 3 florins and 19 stivers (Z twenties, 
and 19) ; putting down 19 we carry 3. 6 times 27 are 1G2, 
and the 3 to be carried are 165 florins. 

This rule is merely the converse of the last. It is evident 
that multiplying by 20 and 12, and dividing the product by 2 
and 20, would give the eamo result. Thus 

£ s. d. 
27 13 2 
20 

568 

. 12 



2)6638 
20)3319 



165fi. IDst, the same result as before. 

EXERCISES. 

59. How many florins and stivers in 30 pounds, 12 
skillings, and 1 grote ? Ans. 183 fl., 12 st., 1 g. 

60. How many florins, &c., in 129 pounds, 7 skil- 
linffs ? Ans. 776 fl. 2 st. 

61. In 97 pounds, 8 skillings, 2 grotes, how many 
florins, &c. : Ans. 584 fl. 9 st. 



ARBITRATIOX OF EXCHANGES. 



283 



QUESTIONS. 

1. What is exchange ? [1]. 

2. What is the difference between real and imagin- 
ary money ? [!]• 

3. What are the par and course of exchange ? [IJ. 

4. Wliat is agio? [3]. 

5. What is the difference between current or cash 
noney and exchange or bank money ? [3] . 

6. How is bank reduced to current money ? [4]. 

7. How is current reduced to bank money ? [6] . 

8. How is foreign reduced to British money ? [7] . 

9. How is British reduced to foreign money ? [8].^ 

10. How are florins, &o., reduced to pounds Flemish, 

11. How are pounds Flemish, &c., reduced to florins, 
fee? [11]. 



ARBITRATION OF EXCHANGES. 
12. In the rule of exchange only two places are con- 
ecfned ; it may sometimes, however, be more beneficial 
«o the merchant to draw through one or more other 
places. The mode of estimating the value of the money 
of any place, not drawn directly, but through one or 
more other places, is called the arUtration of exchanges^^^ 
and is either simph or cortipound. It is " simple " 
when there is only one intermediate place, " compound " 
when there are 7/wre than one. 

All questions in this rule may bu solved by one or 
more proportions. , 

13. Simple Arbitration of Exchanges.— Given the 
course of exchange between each of two places and 
a thu-d, to find the par of exchange between the 

former. , i • x 

Rule.— Make the given sums of money belonging to 
the third place the first and second terms of the propor- 
tion ; and put, as third term, the equivalent of what is 
in the first. The fourth proportional will be the value 
of what is in the second term, in the kind of mQuey 
contained in the third term. 



t 



284 



ARnrrnATioN of exchanges. 



ExAMPLK. — If London c-ohanfoa with T'nrm nf in; .^ 
franc, an.l with Ani.tcnJa.u at 3-1.. njp Vi Z- ul^ wCJ 
ought to bo tho cour.s« of exoha,.:o,Tctt of p2 t; 
Amstonhun that a n.erohant n.ay without loss ron^frca 
L.Midon to Amsterdam through I'aria '? ""cirom 

Df £lV'.\^n ■• ^"^'- ^'/- ^*K^ equivalent, in Klomi«h money, 

Fiefuilh mi!;:;'" "' '''■ ^^''^' ^^^ ^' ^ ^'^'^"^) ^^ 



^^^- • 240- 



'^Tl andlO f ^^'h?''f'^^ ?' «f ^ franc, in Flemish nu.ney. 

that which belongs to the third place; and 34. 8 i/'tho 
given equivalent of £1. ^ ' ^ -Jii. oa. is tno 

It is evident that, 17U. (Flemish) bein- the value of in,/ 

<,i>iitisn;, out lie will not recp vf> ITJ^,; f.n. fi.,*- i v, 

EXERCISES. 

is 1^5 o5' '^"^^^"Se between London and Amstordani 

wh?l;!f'''^"^'' "^^t<^«dto Petorsburgh 5000 ruble. • 

7'lt fo/"^:Tu''7^^^^^^ «nd London 

ih c t oOd per ruble, but between Petcrsburrrh an-l 
Holland It IS at 90^. Plomish per ruble, and Holl d 

Which will be the more advantageous method for Lon- ' 
don be drawn upon-the direct°or the indirect ? Ans 
Jjondon wdl .o-;iin ]e9 n? 1 en^/ :p u ^ 
by way of Holland "» '' '^ '' ""*"^ P"^'""'"^ 

5000 rnblos— ^1041 T?. 17 p ... , r,,-,-- 

but ^1875 Fre«i;h=il^32 1 l^lVlufh.'" ^'^""^" ' 



ARBITRATION OV EXCHANGES. 



385 



14. Compound Arhitrntion of Exchanges. — To find 
what should bo the course of exchange between two 
places, through two or more others,, that it may be on a 
par with the course of exchange between the same two 
places, dircdly — 

IluLE. — Having reduced monies of the same kind to 
the same denomination, consider each course of exchan<»o 
as a ratio ; set down the dift'orent ratios in a vertical 
column, so that the antecedent of the second sliall be 
of the same kind as the consequent of the first, and the 
antecedent of the third, of the same kind as the conse- 
quent of the second — putting down a note of interroga- 
tion for the unknown term of the imperfect ratio. ThcL 
divide the product of the consequents by the product of 
the antecedents, and the quotient will be the value of tho 
given sum if remitted through the intermediate places. 

Compare with this its value as remitted by tho direct 
exchange. 

15. ExAMPLK.— £824 Flemish being due to me at Am- 
sterdam, it is remitted to France at IGrf. Flemish per franc; 
from Franco to "Venice at 300 francs per GO ducats : from 
Venice to Hamburgh at lOOtZ. per ducat ; from Hamburgh 
to Lisbon at 50f/. per 400 rees ; and from Lisbon to England 
at 5.S-. 8^/. sterling per milrce. Shall I gain or lose, and how 
much, tho exchange between England and Amsterdam being 
34i'. 4t/. per XI sterling ? 

\^d. : 1 franc. 
300 francs.: GO ducats. 
1 ducat : 100 pence Flemish. 
50 pence Flemish : 400 roes. 
1000 rees ; G8 pence BritisJi. 
'? : £824 Flemish. 
^XC0xl00x400x68x824 ,.^ 
'10X300X1X50X1000 =^'^ '^"^ ''^'^^'° *^^' **^^^"« 

[Sec. V. 47]) 11^^=£5G0 Gs. A\d. 

But the exchange between England and Amsterdam fd 
£824 Flemish is £480 sterling. 

Since 34s. M. : £824 : : £1 : .^^^'^.^£430. 



I gain therefore by the circular 
minus £480=£80 65. Aid. 



34.S. 4d. 
exchange X5G0 G*-. 4|u. 



286 



AKniTRATION OF RXCnANQES. 



If commission ia chaxf^d in any of the places, it must 
bo do(Juct(!d from tho value of tho sum which cuu bo 
obtained in that place. 

Tho procoss given for tho compound arbitration of ox- 
cliiin;5o may bo provo<l to bo correct, by putting down tbo 
difFerent proportions, and nolving tbcm in Hueeeswion. 'Ibus, 
if 10.'/. aro equal to 1 franc, what will 300 francs (=00 
ducats) bo worth, ff tlio quantity last found is tho valuo of 
00 ducats, what will be that of cue dueat (=3l00t/.), &o. '? 

EXfiKCLSES. 

3. If London would remit iDlOOO sterling to Spain, 
tho direct exchange being 42),(l. per pia.stre of 272 
maravedis ; it i.s ankoA whether it will bo more profit- 
able to remit directly, or to remit first to Holland at 
3o5. per pound ; thence to France at Id^d. per franc ; 
thence to Venice at 300 francs per 60 ducats ; and 
thence to Spain at 3G0 maravedis per ducat ? Ans. 
The circular exchange is more advantageous by 103 
piastres, 3 reals, lOf^- maravcdLs. 

4. A merchant at London has credit for 680 piastres 
at Leghorn, for which ho can draw directly at oOd. per 
pia.stre ; but choosing to try tho cii-cular way, they aro 
by his orders remitted first to Venice at 94 piastres per 
100 ducats; thence to Cadiz at 320 maravedis per 
ducat ; thence to Lisbon at 630 rces per piastre of 272 
maravedis; thence to Amsterdam at 5 W. per crusade 
of 400 rocs ; thence to Paris at IS^d. per franc ; and 
thence to London at 10^^?. per franc ; how mucb is tho 
circular -emittanco better than tho direct draft, reckon- 
ing I per cent, for commission ? Ans. ^£14 12s. l^d 

16. To estimate the gain or loss per cent. — 

lluLE. — Say, as the par of exchange is to the c; urso 

of exchange, so is iElOO to a fourth proportional. From 

this subtract £100. 

Example. — ^The par of exchange is found to be IS^d. 
Flemi.sh, but tho cour.se of exchange is Idd. per fraiic ; 
what is the gain per cent. ? 

£19x100 



lo ia. 



fAOO 



'M 



— =X104 7*-. Ud. 



Thu.s I 

X4 Is. IJ 

If in 
paid, it i 



5. Th 

but tho < 
cent. ? 

6. Th 
course ik 
6*. lli</ 

7. Th 
course of 
Ans. £1 



1. W] 

2. Wl 
pound ai 

3. AVI 

4. Wl 

5. He 
any plac 

6. Ho 



17. T 
gain or 1 
certain ] 

Given 
gain or 1 

KULE 

and at tl 
or loss 

ExAMr 
G(i., and i 

Thetc 
The tc 
Thetc 



I'UOFIT AND LOM. 



287 



Thus (ho piiu por ccnt.=,C104 7s-. l^^ nilnuH £100=* 
X4 7v. 11(/. if the merchant remits through I'liria. 

It' in remitting through Paris oommisHiou must ba 
paid, it is to be deducted from the gnin. 

EXERCISES. 

5. Tho par of exchange is found to bo \8^d. Flemish, 
but the course of exchange is 19|t/., whatis the gain per 
cent. ? Ans. £4 ISs. 2|</. 

6. Tho par of exchange is 17 ^d. Flemish, but tho 
course Is 18|tZ., what is the gain per cent. ? Ans. £4 
6s. UU. 

7. The par of exchange is 18^^. Flemish, but tho 
course of exchange is 17|^rf., what is the loss per cent. ? 
Ans. £1 165. 2d. 

QUESTIONS. 

1. What is meant by arbitration of exchanges .? [12]. 

2. What is the difference between simple and com- 
pound arbitration } [12]. 

3. AVhat is the rule for simple arbitration ? [13]. 

4. What is tho rule for compound aibitration ? [14]. 

5. How arc we to act if commission is charged m 
any place .? [15]. 

6. How is the gain or loss per cent, estimated } [16]. 



PROFIT AND LOSS. 

. 17. This rule enables us to discover how much we 
gain or lose in mercantile transactions, when we sell at 
certain prices. 

Given the prime cost and selling price, to find the 
gain or loss in a certain quantity. 

KuLE. — Find the price of the goods at prime cost 
and at the selling price ; the difference will be the gain 
or loss on a given quantity 

Example. — What do T gain, if I buy 460 lb of butter at 
ijd.j and sell it at Id. per lb ? 

The total prime cost is 460J.x6=2760f?. 

Tlic total sGlliiig price is lOuCi.X i=o^^Od. 

The total gain is o220(/. minus 27G0J.=460c/.=jCl 18s. id. 



r. 



i ♦ 





i i 



288 



PROFIT AND LOSS. 



% 



EXERCISES. 

1. BougLt 140 ft) of butter, at lOd. per ih, and eold 
it nt 14d. por ft) ; what was gained ? Ans. £,'1 6s. 8</. 

2. Bought 5 cwt., 3 qrs., 14 lb of cheese, at £2 I2s. 
per cwt., and sold it for d22 185. per cwt. What was 
the gain upon the ^vholc ? Ans. £1 15s. 3d. 

3. Bought 5 cwt., 3 qrs., 14 ft) of bacon, at 345. per 

What was the 



cwt. 



and sold it at 365. 



4d. per cwt. 
gain on the whole .'' Ans. I3s. 8^d. 

4. If a chest of tea, containing 144 ft) is bought 
for 6s. 8ft. per ft), what is the gain, the price received 
for the whole being £57 10s. } Ans. £9 lOs. 

18. 1*0 find the gain or loss per cent. — 

Rule.: — Say, as the cost is to the selling price, so Is 
£100 to the required sum. The fourth proportional 
minus £100 will be the gain per cent. 

Example 1. — What do I gain per cent, if I buy 1460 lb 

of beef at 3(Z., and sell it at Z^d. per Bb '^ 

3(Z.xl460=4380tf., ia the cost price. 

And 3i(/.xl4G0=5110rf., is the selling nrice. 

5110 X 100 
Then 4380 : 5110 : : 100 : — ^^^ — = £116 13s. 4d. 

Ans. £116 13.S. 4d. minus £100 (=£1^ 135. 4d.) is the gain 
per cent. 

REAijON OF THE RuLE. — The price is to the price plus the 
gain in one case, as the price (£100) is to the price plus the 
gain (£100-f-the gain on £100) in anotiicr. 

Or, the price is to the price plus the gain, as any multiple 
or part of the former (£100 for instamse) is to an equimultiple 
of the latter (£lOO-f-the gain on £100). 

Example 2. — A person sells a horse for £40, and loses 9 

{)er cent., while he should have made 20 per cent. What ia 
lis entire loss "? 

£100 minus the loss, per cent., is 1o £100 as £40 (what 
the horse cost, minus wliat ho lost by it) is to what it cost. 

01 : 100 : : 40 : — — — =£43 19*. liJ., what the horse cost. 

But the person should have gained 20 per cent., or ^ 
of the price j therefore his profit tihould have been 




PROFIT AMI LOSS. 



289 



£ x. d. 

3 19 l.V ia the difference between cost and selling price. 

8 15 9^ is what he should have received above cost. 



12 14 11} is his total loss. 



so IS 



, or \ 
been 



EXERCISES. 

5. Bought beef at 6(Z. per lb, and sold it at ^d. 
What what was the gain per cent. } Ans. 331-. 

6. Bought tea for' 5s. per lb, and sold it for 3s. 
What was the loss per cent. > Ans. 40. 

7. If a pound of tea is bought for Qs. Qd.^ and sold 
for Is. 4d.^ what is the gain per*cent. ? Ans. 12ff . 

8. If 5 cwt., 3 qrs., 26 lb, are bought for £9 85., 
and sold for £11 185. 11^., how much is gained per 
cent. } Ans. 27 ^V^. 

9. When wine is bought at 175, Gd. per gallon, and 
sold for 27.V. 6c/., what is the gain per cent. ? Ans. 57^. 

10. Bought a quantity of goods for j£60, and sold 
them for ^£75 ; what was the gain per cent. .'' Ans. 25. 

^11. Bought a tun of wine for £50, ready money, and 
sold it for £54 IO5., payable in 8 months. How much 
per cent, per amium is gained by that rate .'' Ans. 13^. 

12. Having sold 2 yards of cloth for II5. 6</., I 
gained at the rate of 15 per cent. What would I have 
gained if I had sold it for 12?. t Ans. 20 per cent. 

13. If when I sell cloth at 75. per yard, 1 gain 10 
per cent. ; wh t will I gain per cent, when it is sold for 
85. 6i. .? Ans. £33 Us. 5^d. 

'Is. : 8.S-. 6(!. •: £110 : £133 lis. 5\d. And £133 II5. 
5!/L— £100=£33 il.^ 5 i(Z., is the required gain. 

19. Given the cost price and gain, to find the selling 
price — 

Rule. — Say, as £100 is to £100 plus the gain per 
cent,, so, is the cost price to the required selling price. 

Example. — At what price per yard must I sell 427 yards 
of cloth which I bought for 19*'. per yard, so that I may 
gain 8 per cent. 1 

lOSxiO.N'. 
100 : 108 : : 10s. : — iqq— =JC1 O5. G\d. 

This result may be proved by the last rule. 



290 



rnOFIT AND LOSS. 



EXERCISES. 

14. Bought velvet at 4.?. 8f/. per yard ; at what price 
must I sell it, so as to gaia 12^- per cent. ? Ans. At 
55. 3d. 

15. Bought muslin at 55. per yard ; how must it be 
sold, that I may lose 10 per cent. ? Ans. At 4i*. 6d. 

16. If a tun of brandy costs £40, how must it be 
sold, to gain 6i per cent. ? Ans. For j£42 10a\ 

17. Bought hops at ii4 165. per cwt. ; at what rate 
must they be sold, to lose 15 per cent. .? Ans. For £4 
Is. l\d. 

18. A merchant receives 180 casks of raisins, which 
stand him in \Qs. each, and trucks them against other 
merchandize at 28s. per cwt., by which he finds he has 
gained 25 per cent. ; for what, on an average, did he sell 
each cask ^ Ans. 80 lb, nearly. 

20. Given the gain, or loss per cent., and the selling 
price, to find the cost price — 

Rule. — Say, as JGJIOO plus the gain (or as J3100 minus 
the loss) is to £100, so is the selling to the cost price. 

FiXAMPLB 1. — If I sell 72 K) of tea at (js. per lb, and gain 
9 per cent., what did it cost per Jb ? 

109 : 100 : : 6 : — Jq^=5s. M. 

What produces £109 cost £100 ; therefore what pro- 
duces Os. must, at the same rate, cost bs. Qd. 

Example 2. — A merchant buys 97 casks of butter at 30.«. 
each, and selling the butter at £4 per cwt., makes 20 per 
cent. ; for how much did he buy it per cwt. ? 

30.v.x97=2910s,, is the total price. 
Then 100 : 120 : : 2910 : -^~?-^= 3492s., the 



100 



3492s. 



Belling price. And ~q7)7' \='~£T^ )=43G5, is the number 

of cwt. ; and -,jy-=50]^* lb, is the uvcrage weight of each 
cask. 

lb lb .S. 110 vQ 

Then 50}lj : 112 : : 30 : li"^'^' 






: GO*. 8(/. = £3 65. 



8(i., the required cost price, per cwt. 



FELLOWSHIP. 



291 



EXERCISES. 

,19i. Having sold 12 yards of cloth at 20*. per yard, 
and lost 10 per cent., what was the prime cost? Ans. 
22s. 2ld. 

20. Having sold 12 yards of cloth at 20^. per yard, 
and gained 10 per cent., what was the prime cost .'' Ans. 
1 Si-. 2fjd. 

21. Having sold 12 yards of cloth for £5 14^., and 
gained S per cent., what was the prime cost per yard.? 
Ans. 8,?. 9§r/. 

22. For what did I buy 3 cwt. of sugar, which I 
sold for dE6 3a-., and lost 4 per ceait. } Ans. For £Q 
^s. IJ-^. 

23. For what did I buy 53 yards of cloth, which I 
sold for £25, and gained £b \0s. per cent. } Ans. For 
£23 135. 111(7. 

QUESTIONS. 

1. What is the object of the rule .? [17]. 

2. Given the prime cost and selling price, how is 
the profit or loss found } [17]. 

3. How do we find the profit or loss per cent.? [18]. 

4. Grivcn the prime cost and gain, how is the selling 
price found } [l-Jj. 

5. Given the gain or loss per cent, and selling price, 
how do we find the cost price .? [20] . 



FELLOWSHIP. 

21. This rule enables us, when two or more persona 
aie joined in partnership, to estimate the amount of 
profit or loss which belongs to the share of each. 

h'idlowship is either single (simple) or double (com- 
pound). It is single, or simple fellowship, when tlia 
diflerent stocks have been in trade for the same time. 
It is double, or compound fi-llowsliip, when the difiercnt 
^stoi'ks luive biieu employed for diJJV.reiU times. 

This rule also enal)]es us to esti late how much of a 
bankrupt's stock is to ])e given to each creditor. 



293 



FELLOWSHIP. 



22. Single Felloivship. — Rulr. — Say, as the wliolo 
stock^ is to the whole gain or loss, so is each pr-rson's 
contribution ^o the gain or loss which belongs to him. 

Example.— A put £720 into trade, B £340, and C 
^eOGO ; and they gained Ml by the traffic. What is li'a 
share of it ? 
£ 

720 .. 
340 
960 



2020 : £47 :: £310 



X47X340 

— 2020~~ ^^* 

Each person's gain or loss must evidently be proportionai, 
to his contribution. 



EXERCISES. 

1. B and C buy certain merchandizes, amounting 
to £80, of whicli 13 pays £30, and deSO ; and they 
gain £20. How is it to be divided .? Ans. B £7 10s , 
and £12 10.v. 

2. B and C gain by trade £182 ; B put in £300, 
and £400. What is the gain of each t Ans. B £78, 
and C cii5l04. 

3. 2 persons are to share £100 in the proportions 
Of 2 to B and 1 to C. What is the share of each > 
Am. B £66|, C £33|-. 

4. A merchant failing, owes to B £500, and to 
£900; but has only £1100 to meet these demands. 
How much should each creditor receive ? Atis. B £3924, 
and C £707f ^' 

5. Three merchants load a ship with butter; B 
gives 200 casks, C 300, and D 400 ; but when they are 
at sea it is found necessary to throAV 180 casks over- 
board. How much of this loss should fall to the share 
of each merchant ? Ans. 
60, and D SO. 

6. Three persons are to pay a tax of £100 accord- 
ing to their estates. B's yearly prapcrty is £800, G'a 
£600, and D's £400. How much is eacli person's share ? 



B should lose 40 casks, 



Ans. 



n\. 



is £44:^ C's £33^, and D's £223. 



7. Divide 120 into throe sueh parts as shall be to 
each other as 1, 2, and 3 ? Ans. 20, 40, and GO. 



FELLOWSHIP. 



293 



S. A' ship worth £900 is entirely lost ; } of it be- 
Itmged to 13, J- to C, and the rest to D. What should 
be the loss of each, i3540 being received as insurance ? 
Ans. B £45, G £90, and D £225. 

9. Three persons have gained £1320 ; if B were to 
take £6, C ought to take £4, and D £2. What is each 
person's share ? Ans. B's £660, C's £440. and D's 
£220. 

10. B and C have gained £600 ; of this B is to 
have 10 per cent, more than C. How much will each 
receive .? Ans. B £314f , and C £2854. 

11. Three merchants form a company; B puts in 
£150, and C £260 ; D's share of £62, which they gained, 
comes to £16. How much of the gain belongs to B, 
and how much to C ; and what is D's share of the stock ? 
Ans. B's profit is £16 165. 7j\d., C's £29 3s. 4^^d. ; 
and D put in £142 12s. 2^^c?. 

12. Three persons join ; B and C put in a certain 
stock, and D puts in £1090 ; they gain £110, of which 
B takes £35, and C £29. How much did B and C put 
in ; and what is D's share of the gain ? Ans. B put 
in £829 Gs. ll^J^., C £687 3s. 5i-|J. ; and D's part of 
the profit is £46. 

13. Three farmers hold a farm in common ; one pays 
£97 for his portion, another £79, and the third £100. 
The county cess on the farm amounts to £34 ; what is 
each person's share of it ? Ans. £11 18s. U^^d. ; £9 
14s. 7^^d. ; and £12 6s. 4^^d. 

23. Compound Fellowship. — Rule. — Multiply each 
person's stock by the time during which it has been in 
trade ; and say, as the sum of the products is to the whole 
gain or loss, so is each person's product to his share of 
the gain or loss. 

KxAMPLK.— A contributes £30 for 6 months, B £84 for 
11 months, and C £9G for 8 months; and they lose £14. 
What is C's share of this loss 1 

30 X 6=180 for one month. ) 

84x11=924 for one month. } =£1872 for one month. 



y(3X 8=/U8 tor one mo 



nth. V 



1872 : £14 : : £708 



£14x708 



"1.S72 



_ =£0 Is. 4ld., C's bharo 



ir ; I 




Ji^i 



294 



FELLOVVSiJip. 



i 



^„ ii iuunui , <inu, lor the same reason R'r no ^O'U 

for Uio same time; ami C's -m /"/«« oi =« ^ fu .^"* 

EXERCISES. 

in lio^'^i^""? '"5<^^^°t,s enter into partnership ; B puts 

S is. /n % ^ ?r'^'' ? ^'^ '^'- ^°r ^ months, Vu 
1>^38 105 for 11 months; and they gain £86 16* 

j-^o lus., OS i.37 2s., and D's ^£24 4^ 
T^ in;«■^^^' '''"'^ ^-P'J ^^ ^' *'^^ :^^^^'« ^-^nt of a farm, 
and 1) 50 for the rest of the time. How much of the 
Ind D ^ii ^'''''' ^^^ ' ^'''- ^ ^^^ - ' ^ ^I^t't. 

and In ^^^'^^^?"^t':^' A' I^' «"<i C, enter into partnership, 
iAo wo"- fVT "^^^^-^SQl 13.. 4.Z.'^A's stock 

C's i 2^ '?« ''^'.^ "'^l.^rl' ?'^' ^200' 3 months ; and 
t s, X125, 16 months. What is each person's share of 

131^47 '• ' '' '^^^' ^'' ^^0' ^^^ C'^ ^166 

17. Three persons have received ^£665 interest- B 

^nvf 1^ *t ? ?' ^ ''^''''^^^ 5 ^^^^ "^"«^ is each person's 
i)'si2oo ^''^' -^'^ ^^^^' ^'' ^^^^' ^^ 

trado* f'Zl^v^ '^I'^T ^«.°"ipa°y- X's stock is in 
trade 3 months and he claims J^ of the gain : Y's 
Btock IS 9 months in trade ; and Z advanced^e756 for 
4 months, and claims half the profit. How much did 
X and Y contribute } Ans. X ^£168, and Y £280. 
It follows that Y's gain was A. Then -'- • » • • 4"T=.e.syA . 

pay it60 ; the first sent into it 56 liorses for 12 days, tho 



FELLOWSHIl'. 



295 



Bocond64 for 15 days, and tlio third SO for IS days. 
What must each pay ? Am. The first must pay £17 
10s,, the second £2o, and the third i;37 10a-. 

20. Three merchants are concerned in a steam vessel ; 
the first, A, puts in £240 for 6 months ; the second, ]J, 
a sum_ unknown for 12 months ; and the third, C, ^£160, 
for a time not known when the accounts were settled. A 
received £300 for his stock and profit, B £000 for his, 
and C £200 for liis ; what was B's stock, and O's time ? 
Ans. B's stock was £400 ; and C's time was 15 months. 

If £300 arise from £240 in C months, £000 (B's stock and 
profit) will bo found to arise from £400 (B's stock) in 12 
months. 

Then £400 : £160 :: £200 (the profit on £400 'n 12 
montlis) : £80 (the profit on £100 in 12 months). And £l604- 
80 (£1G0 with its profit for 12 montlis) : £260 (£160 with 
Its profit for some other time) :: J2 (the number of months 
•^ *u s 260x12 , 

in the one case) : j^Xg^ (the number of months in the other 

casc)=]3, the number of months required to produce the 
difterence between £160, C's stock, and the £260, which he 
received. 

21. In the foregoing question A's gain was £60 
during (3 months, li's £200 during 12 months, and C'a 
£100 during 13 months; and the sum of the- products 
of their stocks and times is 8320. What wri(> their 
stocks ? Ans. A's was £240, B's £400, and C's £160. 

22. In the same question the sum of the stocks is 
£800 ; A' stock was in trade 6 months, B's 12 months, 
and C's 15 months; and at the settling of accounts, 
A is paid £60 of the gain, B £200, and C £100. 
What was each person's stock ? Ans. A's was £240, 
B's £400, and C's £160. ' 

QUESTIOiS'S. 

1. What is fellowship .? [21]. 

2. What is the difterence between single and douhle 
fellowship ; and are those ever called })y any other 
names .^ [21]. 

3. What are the rules for single, and double fellow- 
ship .? [22 and 23]. 





' litmmi 






wSi 


il 


II 


'l^H 


m 



296 



BARTKU. 



BAUTER. 

25. Barter enables the merchant to exchange ono 
commodity for another, without either loss or gain. 

lluLE.— Find the price of the given quantity of ono 
kind of merchandise to be bartered ; and then ascertain 
how much of the other kuid tliis price ought to pux- 
. chase. 

ExA>[PT,E 1.— How much tea, at 8s. per lb, ou^ht to be 
t'-ivcn for 3 cwt. of tallow, at £1 10s. Sd. per cwt. 1 
£. s. d. 
1 16 8 
3 



5 



10 is the price of 3 cwt. of tallow. 

And £5 10s.-^8s.=13^, is the number of pounds of tea 
which £o 10s., the price of the tallow, would purchase. 

There must be so many pounds of tea, as will be equal to 
the number of times that 8s. is contained in the price of tho 
tallow. 

E.vAMPi.E 2.— I desire to barter 96 lb of sugar, which 
cost me Sd. per lb, but which I sell at 13rf., giving 9 
months' credit, for calico which another merchant sells for 
lid. per yard, giving months' credit. How much calico 
ouglit 1 to receive l 

I first find at what price I could sell my sugar, were I to 
give the same credit as he does — 

If 9 months give me 5d. profit, what ought 6 months to 
giveT 



9 : 



5 .6X5 30 ._gv/ 
9 ~9~" ' ■ 



Hence, were I to give months' credit, I should charge 
ll»f/. per lb. Next— 

As my selling price is to my buying price, so ought his 
soiling to be to his buying price, both giving the same credit. 

lit : 8 :: 17 :5>^=12.Z. 

Tlie ]irico oi my f?ugar, inuroiuri;, is t?o a <-•"•> "'• '^^^•■•f 
md of his calico, 12r/. per yard. 
Hence "^^^=04, is tho required number of yards. 



BARTER 



297 



EXERCISES. 

1 . A mevcliant lias 1200 stones of tallow, at 2s. 3ld. 
Iho Ktonc ; 13 has 110 tanned hides, weight 3994 lb, at 
b^d. the lb ; and thoy barter at these rates. How much 
ijwney is A to receive of li, along with the hides > Ans. 

£40 ll5. 2hL , , ,^ ^j 

3. A has silk at Ms. per !b ; B has cloth at 12s. 6rf. 
which cost only 10s. the yard. How much must A charge 
for his silk, to make his profit equal to that of B ? Ans. 

17s. 6d. 

3. A has coffee which he barters at lO^Z. the lb more 
than it cost him, against tea which stands ]5 in lOs., 
but which he rates at 12s. Qd. per.tb. How much did 
the coffee cost at first ? Ans. 3s. 4d. 

4. K and L barter. K has cloth worth 8s. the yard, 
which he barters at 9s. Sd. with L, for linen cloth at 
3s. per yard, which is worth only 2s. 7d. Who has the 
advantage ; and how much linen does L give to K, for 
70 yards of his cloth .? Ans. L gives K 215f yards ; 
and L has the advantage. 

f). 1) has five tons of butter, at £2o lOs. per ton, and 
lOi tons of tallow, at £33 15s. per ton, which he barters 
witli ; agreeing to receive i2150 Is. 6d. in ready 
money, and the rest in beef, at 21s. per barrel. How 
many barrels 's he to receive > Ans. 316. 

6. I hi've cloth at Sd. the yard, and in barter charge 
for it at 13^/., and give 9 months' time for payment; 
mo,. ''^ant has goods which cost him 12^. per 
lb, an hich he gives 6 months' time for payment. 
IIow hi : he charge his goods to make an equal 
barter .^^ ...... At 17^^. ,^ , . . 

7. I barter goods which cost 8d. per lb, but tor 
which I charge 13f^., giving 9 months' time, for goods 
which are charged at 17 d., and with which 6 months' 
time are given. Required the cost of what I receive > 
Ans. I2d. 

8. Two persons barter ; A has sugar at Sd. per lb, 
charges it at 13d., and gives 9 months time ; B has 

at 12d. per lb, and charges it at 1 7d. per lb. How 
time must B give, to make the barter equal? 
6 months. 



•f 



298 



ALLIGATION. 



QUESTIONS. 



1. What is barter ? [25]. 

2. Wiiat i.s tlie rule for k 



'arter? [25], 



ALLIGATION. 

t IS called alligation medial; or what in-redients wl 
oe rcQuircd to np/w]i.«,> „ ^ • . ^^o^^uiLms will 

tlioy will produce— 'fertaicnts, to Iind the mixture 

nmnbc,- of tl,e lo«.o.st denomination confined in th^ 
whole ,,„a„„ty, „„d tho qnotient will boX Ate or 

d. d. 

9X08 = 882 

6x87 = 435 

6x34 = 204 . 

219 219)"l52l 

Ans. Id. per ib, nearly. 
The price of each e']"-nr. is fhp nnmV.«« „*• 
multipliod by tho iunn%;r if pou d ami fh^' ^'' P^"?^ 
whole is the mm of tho pricon B t 'if ° IQ ih /'"'"^ "^ ^^'° 
cost lo21./., ono 11.. ov the '^1o[i, !^A, .^^?.^^*'^^"8'^^' ^^'ive 



21UtI 



part of thiH, li 



I piirt of ]621t/ or '-A^'</ ~ ; 



lust cost the 



ALLIGATION. 



299 



KxAMPLK 2.— What will bo tho price iinv II) of a mixtiiro 
ooiitainins !) lb G oz. of ten at 5s. Or/, per lb, 18 lb at (5> 
per lb, and 4() lb 3 oz. at U.s. 4^^/. per lb « 



lb oz. n. 

9 6 at 5 

18 G 

46 3 9 



d. £ s, 

6 per lb= 2 11 
per lb= 5 8 
4iperlb=21 13 



d. 
GJ 





9 



1177 )29 12 G ; 

Ans. 6d. per oz. nearly 



73 
IG 
Il77 ouncoa. 
And Gd. X 10=8.^., is the price per pound. 
In this case, tho lowest denomination beinff outice.M wo 
reduce the whole to ounces ; and having found the price of an 
ounce, wo multiply it by IG, to find that of a pound. 

E.YAMPi E 3.— A goldsmith has 3 lb of p;old 22 carats line, 
and 2 lb 21 carats lino. What will bo tho linoutss of tha 
mixture ? 

In this case the value of each kind of in-rrcdient is iT.n-c 
scnted by a number of carats — 

lbs 

3x22 = GG 
2x21 = 42 



5 



5)108 



Tlie mixture is^^ carats fine. 

EXERCISES. 

1. A vintner mixed 2 gallons of wine, at lis. por 
gallon, with 1 gallon at 124-., 2 gallons at 9^., and 4 
gallons at 85. What is one gallon of tho mixture worth ? 
Ans. 10s. 

2. 17 gallons of ale, at 9d. per gallon, 14 at 7i^., 5 
at 91^/., and 21 at 4ir/., are mixed together. How 
much per gallon is the mixture worth ? Ans. 7j\d. 

3. Having melted together 7 o?.. of gold 22^ carats 
fine, 121 oz. 21 carats fi'no, and 17 oz. 19 carats fine, I 
wish to know the fineness of each ounce of the mixture ? 
Ans. 20|f carats. 

28. Alligation Alternafe. —Given the nature of the 

mixture, and of the ingredients, to find the relative 

amounts of the latter — 

^ KuLE. — Put down the quantities greater than tho 

given mean (each of them connected with the differenco 



r • 



300 



AM.IOATION 



between it and the moan, by tlio Higii — ) in one column ; 
put tlio difforences botwcnn the remaining (luautitiea 
and the moan (eonncctcd with the quantities to which 
they belong, by the sign + ) in a column to the right 
hand of the former. Unite, by a line, amlipliis with souio 
viinus difference ; and then each difference will cxprii.ss 
how much of the quantity, with whoso difference it is 
connected, should be taken to form the required mixture. 
If any difference is connected with more than one 
other difference, it is to be considered as repeated for 
each of the differences with which it is connected ; and 
thef sum of the differences with which it is connected is 
to be taken as the required amount of the (Quantity 
whose difference it is. 

Example 1.— How many pounds of tea, at 5.5. and 8.?. per 
lb, would form a mixture worth 7."?, per tb '? 

Price. Diflerences. Price, i 



1 

S. S. 

The mean=8— 1- 



.V. s. 
-2-f-5=:thc moan. 



1 IS connected with 2s., the difference l)otween the mean 
and 5s. ; hence there must bo 1 lb at 5s. 2 is connocled 
with 1, tlie difference between 8.<?. and the moan ; honco there 
must be 2 lb at 8s. Then 1 lb of tea at 5s. and 2 ib at 8.s-. 
per ib, will form a mixture worth 7s. per lb — as may bo 
proved by the last rule. 

It is evident that any equimultiples of these quantities 
would answer equally well ; hence a great number of answers 
may be given to such a question. 

Example 2.— How much sugar at Od, Id, 5d., and 10'/ , 
will produce sugar at 8>;/. per ib ? 

Prices. Hirt'eronces. Prices. 



The mean= 



d. d. 
9-1- 
10-2- 



d. d 
-3+5 



the mean. 



1 is connected with 1, the difference between Id. and the 
mean ; hence there is to be 1 ib of sugar at Id. per lb. 2 is 
connected with 3, the difference between 5d. and the mean ; 
hcwee there is to be 2 lb at M.. 1 is connected with 1, the 
difTerence between 9 J. and the mean ; hence there is to be 
1 lb at 9f/. And 3 is connected with 2, the difference between 
lOf/. and the mean; hence there are to bo 3 lb at ]0c/. 
per ib. 



AI.MUAMON. 



301 



CoDHcrniontly wo nro to tiiko I lb ut 7>l., and 2 lb at 5</., 
1 tb at ','*/., uud 3 11) ut lOil. If wo exuiuino wliat inixturo 
tUeso will give [27], wo Hhall find it to bo tlio givon moan. 



ExAMJM.K 3.— What quantities of tea at 4s., 6a., Ss. 
0.«. por lb, will pi'oduoo aniixturo worth Ss. ? 
I'l-ices. Dift'ureucei. Tricui, 



S. S. 

1 -f-4=tho mean. 



and 




Tho moan= 

9-4 

3, 1, and 4 aro connected with 1.?., the difforooo between 
4<. juid tho mean ; thorofore wo aro to tako 3 lb -f- 1 lb -}- 4 
lb of tea, at 4s. per lb. 1 ia connected with 3.'?., l.-j,, and 4.s'., 
tho ditferoncos between 8,s\, Gs., and 9s., and tho moan-, 
thoroforo wo aro to take 1 lb of tea at 8s., 1 tb of toa at Ga-., 
and I lb of tea at 9s. por lb. 

Wo Und in this oxampio that 8s., 6s., and 9s. aro all oon- 
nocted with the same 1 j this shows that 1 lb of oaoh will 
be required. 4s-. is oonnoctod with 3, 1, and 4; there nmsU 
bo, therefore, 3-f-l+4 lb of tea at 4s. 

ExAMPMc 4. — How much of anything, at 3s., 4s., 5s., 7?., 
8s., 9s., lis., and 12s. por lb, would form a mixture worth 
Gs- per lb '? 

Pricci. Diirorences. Prlcei, 



Gs 





1 lb at 3r, 2 lb at 4s., 3 lb at 7s., 2 lb at 8s., 3-|-5+6 (14) 
lb at Ss., 1 lb at 9s., 1 lb at Us., and 1 lb at 12s. per lb, will 
form tho required mixture. 

29. Reason ok , the IIule. — The excess of one ingredient 
above the mean is made to counterbalance what the other 
wants of being equal to tho mean. Thus in example 1, 1 lb 
at 5s. per lb gives a deficiency of 2s. : but this is corrected by 
2s. excess in the 2 lb at 8s. per lb. 

In example 2, 1 lb at Id. gives a deficiency of Id., 1 lb at 9^/. 
gives an exce.ss of Irf. ; but the excess of Id. and the deficiency 
of Id. exactly neutralize each other. 

Again, it is evident that 2 lb at 5.Z. and 8 lb at 10^. are 
Worth just as much as 6 lb at 8rf.— that is, Sd. will b« tha 
ftverugo price if w« mix 2 ib afc iiU. with '6 lb at lOd. 



302 



ALLIGATIOiV. 



■, •»«» -#<i^ 







EXERCISES. 

4. How much wine at 8s. 6d. and ds. per gallon will 
make a mixture worth 8s. lOd. per gallon.? Ans. 2 
gallons at Ss. 6d., and 4 gallons at Qs. per gallon. 

5. IIow much tea at 65. and at 3s. Sd. per Il>, will 
make a mixture worth 4s. Ad. per lb .? Am. 8 Sb at 
(is.y and 20 lb at 3s. 8^. per lb. 

6. A merchant has sugar at 5r/., 10^., and I2d. per 
lb. How much of each kind, mixed together, wili be 
worth M. per lb .? Am. 6 lb at 5^., 3 lb at IQd., and 
3 lb at I2d. ' 

7. A merchant has sugar at bd., 10^., 12^., and 16'^ 
per lb. How many lb of each will form a mixture worth 
lU. per lb? Am. 5 lb at bd., 1 lb at 10^., 1 lb at 
12(Z., and 6 lb at 16^. 

8. A grocer has sugar at bd., Id., 12d., and 13d.. 
per K). ; How much of each kind will form a mixture 
worth lOd. per lb .? Am. 3 lb at 5d., 2 lb at 7d.,3fb 
at 12d.j and 5 lb at 13^. 

30. When a given amount of the mixture is required, 
to find the corresponding amounts of the ingredients— 

Rule.— Find the amount of each ingredient by the 
last rule. ^ Then add the amounts together, and say, as 
their sum is to the amount of any one of them, so is the 
required quantity of the mixture to the correspondinff 
amount of that one. 

Example 1.— What must be the amount of tea at 4s. per 
ft, m 736 lb of a mixture worth 5s. per lb, and containing 
tea at bs., 8s., and l)s. per lb ? 

To produce a mixture worth 5s. per lb, we require 8 lb 
at 4s., 1 at 8s., 1 at 6s., and 1 at 9s. per lb. [28]. But all 
ot these, added together, will make 11 lb*, in which there 
are 8 lb at 4s. Therefore 

lb 

8x736 tt> oz. 

=526 4y*y, the. required quantity 



lb 
11 



lb 
8 



m 

736 



11 



of tea at 4s. 

That is. in 736 lb of flm mlx+nro thp"-" 
m. at 'rs. per lb. The amount of each of the other ingre 
dicnts may be found in the same way. 



Triix tJ\^ xju\J lU I,",- 



ALLIGATIOX. 



303 



rrnJi^l^ F 'u}'T' ^^"^ of Syvacu.o, £;ave a certain 
quaali y of p.ld o fom a ciwn; but when he received it, 
suHpectmo; that th« goLl.niith had taken son^e of the gold 
and «upp hed it« place by a b.ser metal, lie co,nmi«simied 
Auh .).ed_(^S the celebrated mathematician of Syracuse, to 
TZ'T^'x "' '^fP'^i'^"/^^^^^ ^vell founded, aril to what 
oxtrnt Archiriiedes was tor some time unsuccessful in his 
resoarches, unti one day, goin- iuto a bath, he rcmark.Kl 
that he displaced a quantity of water equal to his own bulk 
Seeing at once that the same weight of different bodies 
wou.d, Jf "nmcr,sod in water, displace very dilFcrent quan- 
tities of the fluid he exclaimed with delight that he had 
found the desired solution of the problem Taking a mass 
j>t guld equal HI weight to whatwa^ given to the gohlsmith, 
he tound thnt it displaced less water than the crown : which 
t.ieretoro, was made of a lighter, becnn.so a more bulky 
mortal— and, consequently, was an alloij of goM - 

jNow supposing copper to have boen the substance with 
wliich the crown was adulterated, to find its amount- 

J.et the goiu given by Hiero have ww£rhed 1 lb, this 
won d displace about -O.IL' lb of water; 1 lb ^.f copper void 
d.sp ace about -1124 3b of Avater; but let the criwn have 
displaced only -072 it). Then "^ uo\mi .ia\e 

(rold differs from -072, the memi, by— •020 
Copper differs from it by . . -f-O-lO-l", 

,T ., Copper. Di'I'LTPiices. (Jold. 

Hence, ths moan=.=. 1124 -0404 •020-f-052=thc mean. 

Therefore -020 lb of copper and -0404 ib of gold would 
t>r()duce the alloy in the crown. ^ 

l>ut the crown was supposed to weigh 1 ib ; therefore 

•0G04 lb (-020+ -0404) : -0404 lb • • lib • li^Mil"* 

•0G04 
•GG9=-331 lb is 



GC9 ib,_ the quantity of gold. And 1- 



thc quantity of coppc 



EXERCISKS. 



U. A di-usrgist IS desirous of producinir, from medicine 
at '>J'., (^'.v., S.v and 9.-. per 3b, li cwt. of a mixture 
worth 7s per ]b. How much of each kind must he 
use for the purpose r Ans. 28 lb at 5.$., 56 lb at 6s., 
ufv xKf ai, 05., ana 2:i ih at bi'. per ib. 

10. 27 lb of a mixture worth 4s. 4d. per Ib are re- 
qiured. It IS to contain tea at 5^. and at 3s. 6d. per 




304 



ALLIGATION. 



lb. How mucli of each must bo used ? Ans. 15 ft) at 
5i-., and 12 ib at 3.9. 6cL 

11. How much sugar, at Ad., Gd., and Sd. per ]b, 
must there be in 1 cwt. of a mixture worth 7d. per ib } 
Ans. 18|- lb at 4(/., ISf lb at 6d., and 74| lb at 8d. 
per lb. 

12. How much brandy at 123., 135., 145. , and 14a'. 
Gd. per gallon, must there be in one hogshead of a mix- 
ture worth 135. Gd. per gallon > Ans. 18 gals, at 125., 
9 gals, at 135., 9 gals, at 145., and 27 gals, at 145. Gd. 
per gallon. 

31. When the amount of one ingredient is given, to 
find that of any other — 

lluLE. — Say, as the amount of one ingredient (found 
by the rule) is to the^ii-m amount of the same ingredient, 
so is the amount of any other ingredient (found by the 
rule) to the required quantity of "that other. 

Example 1.— 29 lb of tea at As. per lb i« to bo mixed with 
teas at (js., 85., and Ds. per lb, so as to produce what will be 
•Vortli 5.s\ per lb. What quantities must be used '? 

8 Ih of tea at 45., and 1 ib at 6s., 1 lb at 8s., and 1 lb at 
9s., will make a mixture worth 5s. per lb [271. Therefore 

8 ib (the quantity of tea at 4s. per Ib, as found by the rule) . 
29 R) (the given quantity of the same tea) : : 1 lb (the 

quantity of tea at Gs. per ib, as found by the rule) : 221^ ^^ 

8 
rthe quantity of tea at 6s., Avhich corresponds with 29 lb at 
4s. per lb) ==3-^ lb. 

We may in the same manner find what quantities of tea nt 
8s. and 9s. per lb correspond with 29 lb— or i\\Q given amount 
of tea at 4.s. per lb. 

Example 2.— A refiner has 10 ov.. of gold 20 carats fine 
and melts it with 16 oz. 18 carats fine. What must be 
added to make the mixture 22 carais fine ? 

10 oz. of 20 carats fine=10x20 = 200 carats. 

16 oz. of 18 caratii fme=16xl8 = 288 

26 : 1 : : 488 : 18}'[ carats, the 
fineness of the mixture. 

24 — 22=2 carats baser metal, in a mixture 22 carats fine. 

24 — 18f|=5j% carats baser metal, in a mixture 18 JH 
carats fine. 

Then 2 carats : 22 carats : : 5^^^ : 57 j''^ carats of pure 









ALLIOATION. 



305 



fro] (1- required to ohanse 5 ■',- carats baser metal, into a 
mixture 22 carats line. Ikit tliero are already in the mixtura 
1S|:; Ciirats gcl.l; therefore 57^^,— 18j!;:=:i8f!| carats ir<M 
are to, l)e added to every ounce. There are 20 oz.; therefore 
2GXoH|.;=1008 carats of gold are wanting. There are 
L4 carats ^( page 5) in^everyoz. ; therefore 'i;^^ caratsrr^-12 
'" """'" ' ' " ' There will then' he a uiixturo 



oz. of gold must l)e added 
containing 



oz. car. 
10X20 
]()Xl8 
42x24 



car. 

2')0 

288 

1008 



08 : 1 oz. : : 14DG : 22 carats, the required finoness. 

EXERCISES. 

13. How iTiiicli tea at 6s. per lb must be tnixod with 
12 ii) at 3i-. ikl. per il), so that the mixture maybe 
worth •].?. 4d. per lb .? Ans. 4f lb. 

14. How much brass, at I4d. per tb, and pewter, at 
lO^d. per lb, must I melt with 50 lb of copper, at 16V/. 
per lb, so as to make the mixture worth Is. per lb ? 
Ans. 50 lb of bra.s.s, and 200 lb of pewter. 

15. How murdi gold of 21 and 23 carats fine must 
be mixed with 30 oz. of 20 carats fine, so that the mix- 
ture uuiy bo 22 carats fine r A)is. 30 of 21, and 90 
of 23. 

16. How much wine at 7s. r^d.^ at 5.?. 2d., and at 
4s. 2d. per gallon, must be miyxMl with 20 gallons at 
O.v. 8^/. per gallon, to make the mixture worth 6s. per 
gallon r Ans. 44 gallons at 7s. ixL, 16 gallons at oa- 
2d., and 34 gallons at 4i-. 2d. 

QUESTIONS. 

1. What is alligation medial .? [26]. 

2. What is th,^ rule for alligation me lial > [27]. 

3. What is alligjition altoniato : [26 K 

4. Whnt is the rule for Jilligatim alternate } [28]. 

5. What is the rule, v.hon a certain amount of t) <j 
mixture is required .? [30] . 

6. AVhat is the rule, when i\\(^. a; m\\\ 0\ C'l* or moro 
of the ingredients is <^iveu .^ [31]. 



306 



SECTION IX. 

INVOr.UTION AND EVOLUTION, kc. 

1. iNVOLUxroN. — A qnantlty wliicli is the product of 
two or more factors, each of theiu llie same number, is 
termed a power of that number ; and the number, mul- 
tiplied by itself, is said to })0 invclccd. Thus SXoXo 
(:-^125) is a " power of 5 ;" and 125, is 5 " hivolved." 
A power obtains its denomination from the number of 
times the root (or quantity involved) Is taken as a factor. 
Thus 25 (=5X5) is tlie secovd power of 5. — Tlie 
second power of any number is also called its square. ; 
because a square surface, one of M'hose sid' s is expressed 
by the given number, will have its area indicated by the 
second power of that nun ber ; thus a square, 5 inches 
every way, will contain 25 (the S(|uare of 5) square 
inches ; a Sfjuare 5 feet (svery way, will contain 25 
.square foot, &c. 216 (6X0X<)) is "the lliird power of 
6. — The third power of any nundjer is also termed its 
mill ; because a cube, the length of one of avIi ^e sides 
is expressed by the given number, will have ils solid 
contents indicated by the third power e.f that number. 
Thus a cube 6 inches every way, will contain 125 (the 
cube of 5) cubic, or solid inches; a cube 5 feet every 
way, will contain 125 cubic feet, tic. 

2. In place of setting dov/n all the factors, we put 
down only one of them, and mark how often they are 
supposed to be set down by a small figure, which, since 
it poin/s out the number of the factors, is called the 
i7idc.x, or cxpinnanf:. Tlius ^^ is the abbreviation for 
5x5 : — and 2 is th>5 index. 5^ moans 5X5X5X5X5, 
or 5 in the fifth power S"* means 3X3X3X3, or 3 in 
the fourth power. S' moans 8X8X8X8X^X8X8, 
or 8 in the seventh power, &c. 

3. Someti)nes the vinculum [See. IT. 5] is used in con- 
junction with the index ; thus 5-f-'82 means that the sum 
of 5 and 8 is to be raised to the second power — this 



INTOLUTION. 



307 



is very ciIiTerent from 5 ^+8 ° , wlncli means tlic sum of 

the squares of 5 and S : 5 + 8= being 169 ; while 5^ + S'' 
is only 89. 

4. Iq multiplication the multiplier may be considered 
as a species of index. Thus in 187x5, 5 points out 
how often 187 should be set down as an addend ; and 
187X5 is merely an abbreviation for 187+187+187 + 
187+187 [Sec. 11.41]. In 187% 5 points out how 
often 187 should be set down as a factor ; and 187* ig 
an abbreviation for 187X 187X 187X 187x 187 :— that 
is, the " multiplier" tells the number of the addends^ and 
the " index" or " exponent," the number of the factors. 

5. To raise a number to any power — 

Rule. — Find the product of so many factors as the 
index of the proposed power contains units — each of the 
factors being the number which is to be involved. 

Example 1. — What is the 5th power of 7 "? 
7» =7x7x7x7x7=10807. 

Example 2. — What is the amount of £1 afc compound 
interest, for 6 years, allowing G per cent, per annum 1 

The amount of XI for G years, at 6 per cent, is — 
_10GxlOGxl-OGxl-06xlOGxl-06 [Sec. VII. 20], or 
1-00"=1-41852. 

We, as already mentioned [Sec. VII. 23], may abridge 
<\.o process, by using one or more of the products, already 
obtained, as factors. 

■^ EXERCISES. 

1. 3'=243. 

2. 20'"=I0240000000000. 

3. 3^=2187. 

4. 105''=1340095r>40r)25. 

5. 105''=l-3400956-10G25. 

^6. To raise a fraction to any power — ' 
Rule. — Raise both numerator and denominator to 
that power. 

Example. — (f)=^ 









to maliiply it ny itself. But to 
multiply it by itself any nuuil)er of tiinos, we must multiply 
its numerator by itwelf, and also its deuomiuator by itself, ihaf 
number of times [Sec. IV. 00]. 



^os 



EVOLUTION. 



R /•.■^^7 'J Mil 

"• U ^ — 11T38V - 
() / o\n :i I :.';■. 

7. To raise a mixed nunibor to any power — 
lluLE. — llcduoo it to an improper fraction [Sec. IV 
24] ; and then proceed as directed by the lust rple. 

EXAMI'LK.— (21)4=(|)4=fyi^5. 

EXERCISKS. 



10- K-r.J - T.^ 

11. (3^)^=«u^;;^^ 

- - ■■ - .2 2 1 (i-ir 



£9 



8. Evolution is a process exactly opposite to mvolution , 
since, by means of it, v/e find what number, raised to a 
given power, would produce a given quantity — the num- 
ber so fnund is termed a root. Thus wc " evolve " 25 
when wo take, for instance, its square root ; that is, wh-en 
wc find what number, multiplied by itself, will produce 
25, Roots, also, are expressed by e.rjjonenls — but as these 
exponents are fractions, the roots are called ^^ fractional 
powers." Thus 4^ means the square root of 4 ; 4^ the 
cube root of 4 ; and 4^ tlie seventh root of the fifth power 
of 4. Hoots are also expressed by ^, called the radical 
sign. When used alone, it means the square root — thus 
^3, is the square root of 3 ; but other roots are indicated 
by a small figure placed within it — thus ^6 ; which 

means the cube root of 5. ^7^ (7^)? is the cube root 
of the square of 7. 

9. The fractional exponent, and radical sign are some- 
times used in conjunction with the vinculum. Thua 

4—3% is the s quare root of the difierence between 4 

and 3 ; ^o-{-7^ or 5+7'^, is the cube root of the sura 
of 5 and 7. 

iO. To find the square root of any number — 
Rule — I. Point off the digits in pairs, by dots ; put- 
ting one dot over the units' jo/acfi, and then another dot 
over every second digit both to tha right and left of 
the units' place — if there are digits at both sides of the 
decimal point. 




EVOLUTION. 



309 



IT. Find the highest immber the square of which 
will not exceed the amount of the highest period, or 
that which is at tlie- extreuio hift— this number will bo 
the first digit in tho required square root. Subtract its 
square from the highest period, and to the remainder, 
considered as hundreds, add the next period. 

III. Find the highest digit, wliich being multiplied 
into twice the part of the root already found (consi- 
dered as so many tens) , and into itself, tho sum of tho 
products will not exceed the s^tm. of the last remainder 
and tho period added to it. Put this digit in the root 
after the one last found, and subtract the former si07>i 
from the latter. 

IV. To the remainder, last obtained, bring down 
another period, and proceed as before. Continue this 
process until the exact square root, or a sufiicicntly 
noar approximation to it is obtained. 

11. I'LxAMPLK.— What is the square root of 22420225 '^ 

22420225(4735, is the required root. 
1G__ 

87)042 ,-;;■ :"'•■ - 

GOO ,,. . . . • 

943)3302 ■ "^-^ ' • .-'■ 

2820 .-.-■■■ .. • 

0405)47325 
47325 

22 i« tlie highest period; and 4^ is the highest square wlucli 
doo.s not exceed it— we put 4 in tlie root, and subtract 4'-', 
or 10 from 22. This leaves 0, which, along with 42, the next 
poriod. malccs 042. 

We subtract 87 (twice 4 tcns-{-7, the highest digit yhicIi 
wo can now put in the root) X 7 from 042. This loaves 
33, which, along witli 02, the next period, makes 3302. 

We subtract 043 (twice 47 tens -\-'i, the next digit of th(^ 
root) X3 from 3302. This lca.vo.s 473, ^^-jiich, ixhmu V;:t'.^ 
25, the only remaining period, makes 47325, 

We subtract 0405 (twico 473 tons J..'",, the np:.c digit of 
the root) X5. Thi.s leaves n-'^ romaiuder, 

The given numbp,v, therefore, is exactly r. square; and 
its squi\re root is 4735, 

12. llKAsoiv OF I.— Wc point off ';„o .^ip-itg of tlie given 
square in pairs, and consider tlio ^^j,i^{)cr of dots as indicating 



310 



iJVOLUTIOr* 



^^ii^ir^ ' "*■ ''■'='" '■" .""' '■'>"'• ''"'" ""M-or one nor two 

the root—since it will be necessarv fn K^,-^~\5 ^ '^ .^ , 

for each new digit; but Zr"o1hri^e*"w^rn«„?°b:?e°,K™'' 
Keabon or II —We subtract from llie Wcliost Mrio,l of '(!,. 
fml 00?^"'°''.,""' '■'8''''" 'I""' "W«h £ nJt ™co°d U 
dit of I „"■ "'? T' "' *■' "1""° »« the 8rst or hTlc,; 

600 m JZ\, 'i?.'^"""^ by to digits mi. for iustance,"nto 

will contain not only IO2 and 42 hnf nian +!;• li ^^^"^^^ 
rvf in oTiri .1 \u "l/ , , * ' "^^ *'So twice the product 

£Xf ^r r s .t. rsriort^^i'triai 

cedS it "WX'Z^ " ''^ "!° "«' »f tl.e root wl^iSp? .' 
whenwesubracfS7v7 """Pf "''f'' "'•«*■■"*« the rule. 

4000 =16000000 ' 

6420225 

2X4000X700+700^= 6090000 

2X4000X30+2X700X30+30*= 282900 



8X4000X5+2X700X5+2X30X5+5^=17325 



EVOLUTION 



3tl 






of twice tlic sum) of tlio parta of tlio mot nlrc.idy found, 
jnulMphed by tlio ncAV digit, Tims 22420225, the 8quavo of 
4785 contains 4000^-f700--|-30^-f5^ and also Uvica 4000X 
700 + twice 4000X30 4- twice 4000x5; plus twice 700x304- 
twice /00X6; phis twice 80x5:— that is. the square of each 
ot Its parts, with the euui of twice tlio product of every two of 
them (which is the same as each of tliem multiplied by twice 
the sum of all the rest). This would, on examination, be 
lound the case with the square e)f any other number. 

If we examine the cxamjile given, we shall find that it will 
not be necessary to bring down more than one period at a 
time, nor to add cyphers to tlie quantities subtracted. 

13. When the given square contains decimals — 
; If any of the periods consist of decimals, the digits 
m the root obtained on bringing down these periods to 
the remainders will also be decimals. Thus, taking the 
example jus t given, bu t altering th e decimal point, wo 
Bh all have ^2 24202-25= 473- 5; V224 2-0225=47-35. 
^22-42022 5 = 4-735; V^2420225 = -4735 ; and 
^•0022420225 = -04735, &c. : this is obvious. If there 
is an odd number of decimal places in the power, it 
must be made even by the add ition of a cypher. Using 
the same figures, ^2242022-5= 1497-338, &c. 

2242022-56 (1497- 338, &o 



24)124 
_%_ 

289)2820 
2CM_ 

2987)2H)22 
20909 



29943)101350 
89829 



299463)1152100 
898389 _ 

2994668)26371100 
23957344 



1413756 



in this case the highest period consists but of a single digit 

nilU flip frlVf>n linTYlVvisV lO »irif o ^n-,<fnn4- <./>,,.'!«» 

There must be an even number of decimal places ; .lince nc 
number of decimals in the root will produce an odd numbe? 
in thi^ square [Sec. II. 48]— as may be proved by experimen* 



ia_Ji 




312 





EVOH 
KXKR 


JTION. 
CISES. 

20. 
21. 
22. 
23. 
24. 
25. 




14. 
15. 
10. 
17. 


yi95304=442 
^328329— 573 
^•0070= -26 
^87 -05=9 -3022 
^^801=29 -3428 
^984004=992 


^5=2- 23007 
y- 6= -707 100 
V'Ol -9081— 9-59 
.y 238 144=488 


18. 
10. 


^^^2 -3761=5 -09 
^•33 1770= -576 



14. To extract the square root of a fraction — 
EuLE. — Having reduced the fraction to its lowest 
torni.s, make the square root of its numerator the nume- 
rator, and the square root of its donominatcr the deno- 
minator of the required root. 

Example.— y*=f. 

16. Reason of the Rule.— The square root of any quau- 
tity must bo such a number as. multiplied by itself, will pro- 
duce that quantity. Therefore f^ is the square root of | ; for 
I y^ l=ff- ^^e same might be shown by any other example. 

Basides, to square a fraction, we must multiply its numera- 
tor by itself, and its denominator by itself [6] ; therefore, to 
take its square root— that is, to bring back both numerator 
and denominator to what they were before— we must take tbe 
square root of each. 

16. Or, when the numerator and denominator are 
not squares — 

Rule. — Multiply the numerator and denominator 
together ; then make the square root of the product the 
numerator of the require 1 root, and the given denomi- 
nator its denominator ; or make the square root of the 
product the denominator of the requu'ed root, and the 
given numerator its numerator. 

Example.— What is the square root of f ? (|)J a 



=4-472136-{-5='894427. 



^/1X5 ^^ 

6 ^6X4 

17. We, in this case, only multiply the numerator and 
denominator by the same number, and then extract the square 

root of each product. ^^^^ 5=5"^' or ^. Therefore (|)^ 

''4x4 -a 4 



V5X5/ 



2 



_-s/^X5 



—- L--, or 



\5X4/ 



V5X4* 



A> 



EVOLUTION. 



313 



Ifi. Or, lastly— 

lliTLE. — Hediico the given fraction to a decimal 
[Seo IV. 63J, and extract its square root [13J 



EXERCIHKS. 



20 /22\i 28-5300852 



27 



28. 



\37/ "^ 



37 
14 



14 '9000295 
6-244998 



13/ 



13 



29. 



30. 



Sli 



(^|-)^=-745350 
(j^y.=:' 8000254 



(f)'- 



8451542 



19. To extract the square root of a mixed number — 
Rule. — lleduce it to an improper fraction, and then 
proceed as already directed [14, &c.] 



Example.— y2.I=y^ 



=^=H. 



EXERCISES. 



32. y51|j=71 

33. y27VV=5i 

34. yl ''o^lOlSSS 



35. v'lI|=-l-lG83 
3G. y_0,^=2-5298 
37. ^'13^=30332 



20. To find the cube root of any ni;..\ber — 

Rule — I. Point oif the digits in threes, by dots — 
putting the first dot over the units' place., and then 
proceeding boi/i to the right and left hand, if there aro 
digits at both sides of the decimal point. 

II. Find the highest digit whose cube will not ex- 
ceed the highest period, or that which is to the left hanu 
side — this will be the highest digit of the required root; 
subtract its cube, and bring down the next period to 
the remainder. 

HE. Eind the highest digit, which, being multiplied 
by 300 times the square of that part of the root, 
already found — being squared and then multiplied by 
30 times the part of the root already found — and being 
multiplied by its own square — the su7)i of all the pro- 
ducts will not exceed the suvi of the last remainder and 
the period brought down to it. — Put this digit in tho 
root after what is already there, and subtract the former 
ium from the latter. 

IV. To what now remains, bring down the next 



r ■' 



314 



EVOLUTION. 



period, niul procooJ ns botorc. Continue tliia process 
until tlio exact cube root, or a suflicieutly near ajtproxU 
ination to it, is obtained. 

ExAMi'LE.--\Vhat i8 the cube root of 1795970G9288 ? 

179597009288(5042, tho required root. 
125 



300x5»x0 

30x5 xO» 

G'XO 

30()x50»x4 

30x50x4» 

4«x4 

300x5G4'''x2 

30x504x2^ 
2*x2 



545!) 
= 500 



3981009 
3790144 

190925288 
190925288 



We find (by trial) tliat 5 is tho first, the second, 4 tho 
third, and 2 tlio last digit of tho root. And the given 
number is exactly a cube. 

21. IIeason of I. — We point off the digits in threes, for a 
reason similar to that which caused us to point thorn off in 
tf?os, when extracting the square root [12]. 

Reason of II. — Each cube will be found to contain the 
cube of each part of its cube root. 

Reasoist of III. — The cube of a number divided into any 
two parts, will be found to contaiu, besides the sum of the 
cubes of its parts, tlie sum of 3 times the product of «ach 
part by tlie otl.er part, and 3 times the product of each vart 
by the squaio of tho other part. This will appear from the 
following : — 

179597069288 
5000*=1 25000000000 



54597069288 
X 5000"' X GOO-f 3 X 5000 X G00'-}-G00*= 5061 6000000 



3 X 5000' X 40-f 3 X 5G00 X 40^+40' 



3981009288 
3790144000 



190925288 
8 X 5640^ X 2 -(-8 X 5640 X 2*+2'= 1 90925288 

Hence, to find the second digit of the root, we must find by 
tnai some rrarnbcr which — being multiplied hj 3 times the 
square of the part of the root already found — its square being 



EVOLUTION. 



315 



mnltiplio'l l»y H Hmoa tlio part of tho root nlromly fotin.l— and 
lii'itig iimltiplicd by tho nqunrc of UhoU'— tho Htim of the pro- 
ducts will not exceed wliat rornnins of tho j^fiven numhnr. 

JiiHtoiuI of couHideriiif^ tlio part of tlio rnot iilretidy fdund ns 
to many tens [i2J of the denoiuiiiatiou next fdllowiiig (jih it 
rt'iilly Ih), which woidd (idd one cypher to it, ami two cyphers 
to ItH aquaro, wo consider it as so many iinitH, and multiply 
It, not )>y 3, but by HO, ami its Bquarc, not by %, but by 800. 
For 800 X 5' X t5 -i- '"'^ X & X G'-f-G'X') Ih the sanio thing as 
8xr>0'XG-f-3x50Xt»'+<)'X'»; since Ave only change tho posi- 
tion of the factora 100 and 10, which docs not alter tho product 
[Sect. 11. 35]. 

It in evidently unncccHsnry to bring* down more than ono 
period at a time ; or to add cypherB to tlie subtraliendH. 

Ukasov ok IV. — The portion of tlie root already fniind may 
be treated as if it "ro a sinfflo digit. 8inco into wliatever 
two parts wo («livido any number, its cube root will contain 
tlio cube of ench part, with o times the flquaro of each multi- 
plied into the other. 

22. Whon there me. decimals in tho given cube — 

If any of the periods consist of decimals, it is evident 
that the difji;its found on bringinjjj down tbeso periods 
Ljust be decimals. Thus ^17U.5'}7-()6928S = 5n-42, &c. 

When the dtMJimals do not form complete periods, the 
periods are to bo completed by the addition of cyphers. 

ExABiPLE. — What is tho cube root of -3 '? 



0'800(-CG9, &c. 
21G 



800X6'X6 
SOXGXG^ 
GXG' 

800 X 66' X 9 
80XGGX9' 
0X9* 
•669, &c. And 



84000 
=71496 



12504000 
=11922309 



581G91, &c. 

^•3='669, &c. And -3 is not exactly a cube. 

It is ncce.ssary, in this case, to add cyphers; since ono decimal 
in the root will give 3 decimal places in the cube; two decimal 
^laces in the root will give six in the cube, &c. [Sec. II. 48.] 



KXKRCISES. 



88. yp=3- 207534 

89. 4/39=3 -391211 

40. y2r2=5-962731 

41 . ^n 23505'.!92=4 98 

42. ^190r0U37"5=575 



43. ;/458ai4011=771 

44. ^ 483 • 736 (325=^7 -85 

45. ^•G3G05a=-86 
4(^ 3/099=') •ODGGGG 



47. y- 979140657= -993 



i! 





1 




i 


1 



31G 



EVOLUTION. 



.2 



23. To extract the cube root of a fraction — 
3luLE. — JIaving reduced the giveu fraction to its 
lowest terms, make tlie cube root of its numerator the 
luimerator of the required fraction, and the cube root 
of its denominator, the deuomiuator. 

^''' ^125 

21 Reason- of the Rule.— The cube root of any number 
must be such as that, taken three times as a factor, it will 
procluce that number. Tlierelbre f is the cube root of - 3^^; 
fov j X I X f = yI ^.— Tlie same thing might be shown, by uuv 
otiier example. '' 

Resides, to cube a fraction, we must cube both numerator au<? 
denominator; therefore, to take its cube root— tliat is to reduce 
It to what it was before— wo must take the cube root of both. 

25. Or, when the numerator and denominator are 
not cubes — 

llui.E. — IMuItiply the numerator by the square of tlu^ 
denominator ; and then divide the cube root of the pro- 
duct by the given denominator; or divide tlie given 
numerator by the cube root of the product of the given 
denominator multiplied by the square of the giveu 
numerator. 

Example.— What is the cube root of 5 ? 
^— 2 

or -.^. = 5-277032 -^ 7 == 753047. 



(./ = ^3XP 



-5/7x3' 
This vale depends on a principle already explained [IG]. 

26. Or, lastly— 

Rule. — lloduce the given fraction to a decimal 
[Sec. IV. 63], and extract its cube root [22] . 



48. 
40. 

50. 



8-G5349( 



\11/ ~5- 604079 



EXEnCISE.S. 

61. 
52. 



7-(>51725 



(|^y=-560907 



■472103 



27. To fijid the cube root of a mixed number — 
lluLE. — Iteduce it to :m improper fraction ; and then 
proceed as already directed [i>3, &c.] 



EVOLUTION. 



317 



EXERCISES. 



54. (28ni=3-0G35 

55. (7})J=l-93098 

56. (9^)i=20928 



57. (71f)*=41553 

58. (32/y)^=31987 

59. (5|)Ul-7592 



28. To extract any root wliatever — 

liuLE. — When the index of the root is some power 
of 2, extract the square root, when it is some power of 3, 
extract the cube root o* the given number so many times, 
Buccessively, as that power of 2, or 3 contains unity. 



/ 



Example 1.— The 8th root of 65530=>/Vy 65536=4, 
Since 8 is the third power of 2, we are to extract the 
square root three times, successively. 



Example 2.— 134217728«=yVl342lT7S=8. 

Since 9 is the second power of 3, we are to extract the 
cube root twice, suocessively. 

29. In other cases we may use the following (Hutton 
Mathemat. Diet. vol. i. p. 135). 

Rule. — Find, by trial, some number which, raised 
to the power indicated by the index of the given root, 
will not be far from the given number. Then say, 
as one less than the index of the root, multiplied by the 
given number — plus one more than the index of the root, 
multiplied by the assumed number raised to tlie power 
expressed by the index of the root : one more than the 
index of the root, multiplied by the given number — 
plus one less than the index of the root, multiplied by 
the assumed number raised to the power indicated by 
the index of the root, : : the assumed root : a ^ still 
nearer approximation. Treat the fourth proportional 
thus obtained in the same way as the assumed number 
was treated, and a still nearer approximation will be 
found. Proceed thus until an approximation as near as 
desirable is discovered. 

Example.— Wliat is the 13th root of 923 1 
Let 2 bo the assumed root, and the proportion will be 
12x923+14x2^-' : 14x923+12x2*^ :: 2 : a nearer 
approximation. Substituting this nearer approximation for 
2, in the above proportion, we get another approximation, 
which wo may treat iu the same way. 



318 



EVOLUTION 



EXKRCISK3. 



GO, (9GG98)K=G-7749 



Gl. (GG457)iT=27-l42 
62. (23G5)?=31-585 



68. (8742G)?=5084-29 

04. (8-9G5)'=l-368 

65. (•07542G)t4=-04G988 



30. To find the squares and cubes, the square and 
cube roots of numbers, by means of the table at the end 
of the treatise — 

This table contains the squares and cubes, the square 
Rnd cube roots of all numbers which do not exceed 1000 
hut it will be found of considerable utility even when very 
hi£':h numbers are concerned — provided the pupil bears 
in^inind that [12] the square of ai\y number is equal to 
the sum of the squares of its parts (which may be found 
by the table) plus twice the product of each part by the 
sura of all the others ; and that [2 1 ] the cube of a 
number divided into any two parts is equal to the sum 
of the cubes of its parts (which may be found by the 
table) plus three times tne product of each part multi- 
plied by the square (found by means of the table) of 
Hie other. One or two illustrations will render this 
sufficiently clear. 

Example 1. — Find the square of S734r)G. 

873450 maybe divided into two parts, 873 (thousand) and 
45G (units) . But we find by the table that 873'=7G2120 and 
450'=20793G. 

Therefore 762129000000=873000' 

700176000=873000 X twice 45G 

207936=450' 



And 702025383936=873456' 
ExAMPLK 2.— Find the cube of 864379. Dividint; this into 
864 (thousand )_and 379 (units), wejfind 86?=(vi4972544 
b'64 =746496, 379 =54439939, and 379 =143641 

Therefore 644972544000000000=8(HOOO' 

848765952000000=3 X 804 W X 379 
3723 1 7472000=3 x 804000 x 3?j' 



54439931 



379 



And G45821G82323911939=r-86^ 



LOGARITHMS. 



319 



<5l In finding tlie square and cube roots of larger numbers, 
we obtain their three highest digits at once, if we look in the 
table for the Jiighest cube or square, the highest period of 
which (the required cyphers being added) does not exceed the 
hio-hest period of the given number. The remainder of the 
process, also, may often be greatly abbreviated by means of 
ithe table. • 

QUESTIONS. 

1. What are involution and evolution } [1]. 

2. What are a power, index, and exponent > [1 & 2J. 

3. What is the meaning of square and cube, of the 
B(iuare and cube roots } [I and 8J. 

4. What is the difference between an integral and a 
fractional index .? [2 and 8] . 

5. How is a number raised to any power } [5]. 

6. What is the rule for finding the square root } [10]. 

7. What is the rule for finding the cube root ? [20] . 

8. How is the square or cube root of a fraction or 
of a mixed number found > [14, &c., 19, 23, &c., 27]. 

9. How is any root found } [28 and 29] . 

10. How are the squares and cubes, the square roots 
and cube roots, of numbers found, by the table .? [30] . 



LOGARTIHMS. 

32. Logarithms are a set of artificial numbers, which 
reprcsent°the ordinary or 'natural numbers. Taken 
along with what is called the base of the system to 
which they belong, they are the equals of the corres- 
ponding natural numbers, but without it, they are 
merely their representatives. Since the base is un- 
changeable, it is not written along with tlie logarithm. 
The logarithm of any number is that power of the base 
which ts equni to it. Thus 10^ is eqital to 100 ; 10 is 
the hase^ 2 (the index) is the logarithm^ and 100 is the 
corresponding natural number.— Logarithms, therefore, 
are merely the indices which designate certain powers 

of some base. 

33.. Logarithms afford peculiar facilities for calcu- 
lation. For, as we shall sec presently, the multiplica- 
tion of numbers is performed by the addition of their 




320 



LOGARITHMS. 



logarithms ; one number is divided by another if we 
subtract the logarithm of the divisor from that of the 
dividend ; numbers are iuvolveu J' we multiply tJioir 
logarithms by the index of the proposed power ; and 
evolved if wo divide their logarithms by the index of 
tho proposed root.— But it is evident that addition and 
subtraction are much easier than multiplication and 
division ; and that multiplication and division (particu- 
larly when the multipliers and divisors are very small) 
are much easier than involution and evolution. 

34. To use the properties of logarithms, they must bo 
exponents of the same base— that is, the quantities raised 
to those powers which they indicate must be the same. 
Ihus 104X123 is neither 10^ nor 12% the former bein-. 
too small, the latter too great. If, therefore, we desirS 
to multiply 104 and 12« by means of indim, we must 
Imd .some power of 10 which will be equal to 123 or 
some power of 12 which will be equal to 10% or finally 
two powers of some other number which will be equal 
respectively to 10^ and 123, ^^^ then, adding these 
powers of the same number, we shall have that power 
ot It which will represent the product of 10^ and 123 
Ihis explains the necessity for a table of loo-arithms— 
we are obliged to find the powers of some one base which 
will be either equal to all possible numbers, or so neariy 
equal that the inaccuracy is not deserving of notice The 
base of the ordinary system is 19 ; but it is clear that 
tiiere may be as many difierent systems of logarithms 
as there are difi-erent bases, that is, as there are difierent 
numbers. 

35. In the ordinary system— which has been calcu- 
lated with great care, and with enormous labour, 1 is 
the logarithm of 10 ; 2 that of 100 ; 3 that of 1000, &c 
And, since to divide numbers by mejins of these loga- 
nthms (as wo shall find presently), we are to subtract 
the logarithm of the divisor from that of the dividend, 
IS the logarithm of 1, for 1=L^— 10'-'— 10" • — 1 Ls 

10 ' 

the logarithm of -1, for •l=si=lo''=10«— 1^=^10- 



10 101 



for the same reason, -2 is the logarithm of -01 : 
that ol -001, &c. ' 



-1 . „„.! 

, uuu 



bcr, a 



LOOAUITHMS. 



521 



^■', or 



'3 



36. The logai-itlinis of numbers hetwem 1 
must 'be more tlifin and less than 1 ; that is, 
Bomo decimal. 



and 1 0, 
, Kn.iu xn, must bo 
The logarithms of numbers between JO 
and 100 must be more than 1, and less than 2 ; that 
is, unity with some decimal, &c. ; and the logarithms of 
numbers between -1 and -Ql must be —1 and"' some deci- 
mal ; between -01 and -001, —2 and some decimal, &c. 
The decimal part of a logarithm is ahoays positive. 

37. As the integral part or charaderisik of a posi- 
tive logarithm is so easily found — being [35] one less 
than tlie number of integers in its corresponding num* 
bcr, and of a negative logarithm one more than thu 
number of cyphers prefixed in its natural nuniber, 
it is not set down in the tables. Thus the logarithm 
corresponding to the digits 9872 (that is, its decimal 
part) is 99440:'^ ; hence, the logarithm of 9872 is 3 
•994405 ; that of 9S7-2 is 2-994405 ; that of 9-872 is 
0-994405 ; that of -9872 is- 1-994405 (since there is no 
integer, nor prefixed cypher) ; of -009872— 3*994405, 
&c. : — The same digits, whatever may be their value, 
have i\\Q E-AmQ decimals in their logarithms; since it 
is the integral part, only, which changes. Thus the 
logarithm of 57864000 is 7-702408 ; that of 57864, is 
4-7G2408 ; and that of -0000057864, is— 6-762408. 

38. To find the logarithm of a given number, by the 
table — 

Tlie integral part, or charaGtoristic, of the logarithm 
may be found at once, fioni v/liat has been just said [37] — 

When the number is not greater than 100, it will bo 
found in tlie column at the top of which is N, and the 
decimal part of its logarithm iinmediately opposite to it 
in the next column to tlio light liand. 

If the number is greater than 100, and less than 
1000, it will also bo found in the column marked N, 
and the- decimal part of its logarithm opposite to it, iu 
the column at tlie top of which is 0. 

If the number contains 4 digits, the first three of 
them will be found in the column under N, and th« 
fourth at the top of the pngo ; and tlion its logarithm 
in Jic same horizontal lino as the thi(!o first digits of 
the given nunibor, and in the same column as its fourth 



ill 



v 



322 



LOGARITHMS. 



^ 



If the number contains more than 4 digits, find the 
logarilhui of its first, four, and ;ilso the diifcrence be- 
tween that and the h)gaiithni of tiic next higher num- 
ber, in the table ; multiply this diiFereuue by the remain- 
ing digits, and cutting off from the pr(^uct so many 
digits as were in the multiplier (but at the same time 
ftdding unity if the highest cut off is not less than 5), add 
it to thcr logarithm corresponding to the four first digits. 

Example 1.— The logarithm of 59 is !• 770852 (the charac- 
teristic being positive, and 07w less than the number oiintegers) . 

Example 2.— The logarithm of 338 is 2528917. 

Example 3.— The logarithm of -0004587 is — 4(561529 
(tlie characteristic being negative, and one moix than the 
number of prefixed cyphers) . 

Example 4.— The logarithm of 28434 is 4-453838. 

For, the difference between 453777 the logarithm of 2843, 
the four first digits of the given number, and 453930 the 
logarithm of 2844, the next number, is 153 ; which, multi- 
plied by 4, the remaining digit of the given number, pro- 
duces G12: then cutting off one digit from this (since we 
have multiplied by only one digit) it becomes Gl, which being 
added to 453777 (the logaritlim of 2844) makes 453838, and, 
with the characteristic, 4453838, the required logarithm. 

Example 5.— The logarithm of 873457 is 5-941242. 

For, the difference between the logarithms of 8734 and 
8735 is 50, which, being multiplied by 57, the remaining 
digits of the given number, makes 2850; from this we cut 
off two digits to the right (since we have multiplied by two 
digits), when it becomes 28 ; but as the highest digit cut 
off is 5, we add unity, which makes 29. Then 5-941213 (the 
logarithm of 8734) -[-29=5-941242, is the required logarithm. 

39. Except when the logarithms increase very ra- 
pidly — that is, at the commencement of the table — the 
differences may be taken from the right hand column 
(and opposite the three first digits of the given number) 
where the mean differences will be found. 

Instead of multiplying the mean difference by the 
remaining digits (the fifth, &c., to the right) of the given 
number, and cutting off so many places frc^ the product 
as are equal to the number of digits in the multiplier, 
tx) obtain the iir-)pur!luaal part — or what is to be added 



5, 2u. 



LOGARITHMS. 



323 



to the logaritlim of tlie first four digits, we may tako 
the ^-oportioiuil part corrcspouding to each of the re- 
inaitiing digits from that part of the columu at the left 
hand side of the page, which is in tlio same horizontal 
division as tliat in which the first three digits of the 
givvon number have been found. 

K.VAMi'Li:.— What is the logarithm of 839785 ? 

The (decimal part of the) logarithm of 8B9700 is 924124. 
Opposite to 8, in the same horizontal division of the page, 
wo lind 42, or rather, (since it ia 80) 420, and opposite to 
5, 2u. Monce the re(iuirod logarithm'is 9241244-420-f2G=«: 
Vi24570: uud, with the characteristic, 5-924570. 

40. Tlic mctliod given for finding tlie proportional part — or 
what is to bo added to the next lower logarithm, in the table— 
iirisos from tlie diiferoiico of numbers being proportional to the 
ditference of their logarithms. Hence, using the last example, 

100 : 86 : : 62 (92417*i, the logarithm of 839800—924124, 

the logarithm of 839700) : ""Vqq-. or the difference (the ynean 

difference mnj generally be used) X by the remaming digits of 
tlie given nuiMl)or — 100 (the division being performed by cut- 
ting off two digits to the riglit). It is evident that the number 
of (iigits to be cut off depends on the nuniber of digits in the 
multiplier. The logaritlim found is not exactly correct, be- 
cause numbers are not exactly proportional to the difforcncca 
of their logarithms. 

The proportional parts set down in the left hand column, 
have been calculated by making the necessary multiplica- 
tions and divisions. 

41. To find the logaritjiini of a fraction — 

lluLE. — Find the logarithms of both numerator and 
denominator, and tlien subtract the former from tho 
latter ; this will give the logarithm of the quotient. 

Example.— Log.' i| is 1-672098 - 1-748187 = - 1-923910. 
Wo find that 2 is to be subtracted from 1 (the character- 
istic of the numerator) ; l)nt 2 from 1 leaves 1 still to bo 
snbtractcd, or [Sect. II. 15 j — 1, the characteristic of tho 
quotient. 

Wo shall find presently tliat to divide one quantity bj 
anoth.or, avo have merely to subtract the lop;arithm of the I'attei 
iVoni that of tho former. 

42. To find the logarithm of a mixed number — 
iluT.K. — llcduee it to an improper fraction, and pro 

c<"od as directed by the last rule. 



|i| 




/'■ 



324 



LOOARITIIMS. 



43. To fiud the numjicr which corresponds to a given 
logarithm — • '"• 

If the logarithm itself is found in the tahle — 
lluLE. — Take from the table the number which cor- 
responds to it, and place the decimal point so that there 
may be the requisite number of integral, or decimal 
places — according to the characteristic [37]. 

Fa' AMPLE. — What number corresponds to the logarithm 
4-214314? 

AVo find 21 opposite the natural number 103 ; and look- 
i -g along the horizontal line, we find the rest of the logarithm 
under the figure 8 nt the top of the page : therefore the digits 
of the required number are 1038. But as the charaeteribtic 
is 4. there must in it be 5 places of integers. Hence the 
required number is 1G380. 

44. ,If the given logarithm is not found in the table — ■ 
liuLE. — Find that logarithm in the table which is 

next lower than the given one, and its digits will bo 
the highest digits of the required number ; find tho 
diflerence between this logarithm and the given one, 
annex to it a ryphcr, and then divide it by that differ- 
ence in the table, which corresponds to the four highest 
digits of the required number — the quotient will be the 
next digit ; add another cypher, divide again by the 
tabular diflerence, and the quotient will be th-: next 
digit. Continue this process as long as necessary. 

]'2xAi\rPLE. — What number corresponds to the logarithm 
5054329 1 

C54273, which corresponds with the natural number 4511, 
is the logarithm next less than the given one ; therefore the 
first four digits of the required number are 4511. Adding 
a cypher to 50, the difference between 054273 and the given 
logarithm, it becomes 500, which, being divided by 90, tlie 
kihidar difference corresponding with 4511, gives 5 as quo- 
tient, and 80 as remainder, I'herefore, the first five digits 
of the required number are 45115. Adding a cyplier to 80, 
it becomes 800; and, dividing this by 90, we obtain 8 as 
the next digit of the required number, and 32 as remainder, 
"^riio iiife'rcrs of the required numl^or (one more than 5, tho 
characteristic) are, tlierofore, 451158. We may obtain the 
decimals, by continuing the addition of cyphers to the re- 
mainders, and the division by 90. 



4o. 



V 



X. 



LOQARITIIMa. 



325 



45. Wc arrive at tlio same lesult, by Bubtracting 
from .the difference between the given logarithm and 
the next less in the table, the highest (which doesno 
exceed it) of those proportional parts found at the right 
W side^f the page and in the same honzontaWm- 
sion with the first three digits of the given number-- 
continuing the process by the addition of cyphers, until 
nothing, or almost nothing, remains. 

FxAMPLK.-Usin- the last, 4511 is the natural number 
cor esTonUng to the logarithm G54273, which differs from 
he ^enlolarithmby'sG. The Pvop-t-f.r^^^^^ %' 
B-imo horizontal division as 4511, are 10, 19, 2J, c5», 4», oo, 
G7 77 md 80. The highest of those, contained in 56, is 
48. w I'ich we find opposit^e to, and therefore corresponding 
with tho natural number 5; hence 5 is the next of the 
•0 ulred digits. 48 subtracted from 5G, leaves 8 ; this, when 
a'^^pher is^dded, becomes 80, which contains 77 ^corres- 
w>udino- to the natural number 8)5 therefore 8 is the next 
^ the "required digits. 77 subtracted f-m ^O, kaves 3 
tliiK when a cypher is added, becomes 30, &c. ^^^o inte 
to 5 Therefore, of the required number, are found to be 
451158, the sauie as those obtained by the other method. 

The rules for finding the numbers corresponding to 
civon loo-arithms are merely the converse of those used 
for finding the logarithms of given numbers. 

Use of Logarithms in Arithmtic. 

46. To multiply numbers, by means of their loga- 

" Kml-Add the logarithms of the factors ; and the 
natural number corresponding to the result will be the 
required product. 

ExAMPLF..-87x24=1939519 (the log. of 87) -f 1-380211 
Ohe^olof 24)=3319730; which i^fo^^^T ° ^rXx 
ivith the natural number, 2088. Therefore 87x24=2(588. 

from the very nature of indices. Thus f X° — ^0-'^,?^'^O^'^ 
jrom uie vLi^y and the abbreviation for 

multii.liodSXoXoXoXoXoxoxJ, -I indices 

(logatulnlv "The rule rnighl in the same way, be proved 
correct by any other example. 



p Q 







I 



326 



L0CiAlUTJIM8 



47. When tho clmractcristies of tlio logarithms^ to be 
added arc both i)ositivo, it is cvidont that their sum will 
bo positive. When thoy are both negative, their sum 
(diminished by wliat is to bo carried from tlie sum of 
th.i positive [36] decimal parts) will be negative. When 
one is negative, and tho other positive, subtract tho less 
from the greater, and prefix to the difference the Bign 
belonging to tho greater — bearing in mind what has 
been already said [Sec. II. 15] with reference to the 
subtraction of a greater from a less quantity. 

48. To divide numbers, by means of their logarithms — 
liui.R. — Subtract tho logarithm of the divisor from 

that of the dividend ; and tho natural number, corres- 
ponding to tho result, will be tho required quotient. 

Example.— 1134 -f. 42=3054013 (the 1o;t. of 1184) — 
1G23249 (the log. of 42) = 1-431304, which is found to 
correspond with the natural number, 27. Therefore 1134— 
42=27. 

Reasox of the Rule. — This mode of division arises from 
the nature of indices. Thus 4*-i-4'=[2] 4x4X4X4X4— IX 

4X4X4X4X4 ^ ^ 4x4x4 ^ 
4X4= 4>^4x4 — =4X4X4^-^1=4x4, the abbreviation 

for which is 4»; Bu*, 2 is equal to tho index (logarithm) of 
the dividend minus hat of the divisor. Tlie rule might, in 
the same way, be provvd correct by any other example. 

49. In subtracting tho logarithm of the divisor, if it 
is negative, change the sign of its characteristic or inte- 
gral part, and then proceed as if this were to be added 
to the characteristic of tho dividend ; but before making 
the characteristic of the divisor positive, subtract what 
was borrowed^ (if any thing), in subtracting its decimal 
part.^ For, since the decimal part of a logarithm is 
positive, what is borrowed^ in order to make it possible 
to subtvact the decimal part of the logarithm of the 
divisor from that of the dividend, must be so much 
taken away from what is positive, or added to what '3 
negative in the remainder. 

We chac^ge the sipin of tho negative characteristic, ana 
then add it; for, adding a positive, is the same as taking 
awny a negative quantity. 



V 



Ill 



i,o«AurrHM«. 



fsrr 



in 



^0. To -also a quantity to any power, by means of 
it»- logaritli.n — 

lluLE. — Multiply tho lofrarltlim of tlie qiianity by 
tho index Oi t\w power; and the natural number cor- 
rcbponding to the re.sult will be the required power. 

KxAMPi.E. — Puufic 5 to tho 5th power. 

The lo^'arithm of 5 is OO'.IHOT, whieh, niultlpllcd by 5, 
gives 3'4'J486, the logarithm of 3125. Theroforo, the 5th 
jiowcr of 5' is 3125. 

kr.AHON OK Tiiii Rui.K. — Tliis vnlc also follows from tlie 
nawire of imlicea. o* vnisod to tlie otli power is 6X& inuUi])li»>d 
hy Hx'o )uulti{)lieil by oX5 nuillii;licil by SX-) nmltiplieil 
by rjX5, or r)X'''X5xr)X''iX'''>Xi)X'^X':>X0, the abbreviation 
for which is [2] 5'". lUit 10 is eiiiial to 2, the index (b.garithm) 
of tho quantity, niultiplioa by 5, that of the ])Ower. The 
rnle might, in the same way, bo proved correct by uuy other 
example. 

51. It follows from what has been said [47] tbitt wbcu 
a negative chariieteristic is to be multiplied, the produet 
in nrrrativc ; and that what i:< to be carried from the 
nmltijljcation of the decimal part (always positive) is 
to be suhtraclcd from tlii.s mj^ativo result. 

52. To evolve any quantity, by means of its loga- 
rithm — 

Bulk. — Divide the logarithm of the given quantity 
by that number which expresses the root to be taken ; 
and the natural number corresponding to the result will 
be the re(|uired root. 

Ex.ypi.E.— What is the 4th root of 2401. 

Tlie lo;i;arithm of 2401 is 3 o80302, which, divided by 4, 
the number expressing the root, gives -845098, the logarithm 
of 7. Therefore, the fourth root of 2401 is 7, 

Heason ok the Rule. — This vule follows, likewise, from 
tho nature of indices. Thus the 5th root of ItV ia such a 
number as, raised to the 5th power— tliat is, taken 5 times aa 
<i factor— would produce 16'". But loV, taken 5 times as a 
factor, would produce 10'". The rnle might be prove 1 correct, 
equally well, by any other example. 

53. "When a negative characteristic is to bo divided— 
UiTi.i: I.— If the cliaractevistic is cKadly divi.sible by 
the divisor, divide in the ordinary way, but make tho 
characteristic of the quotient negative. 



I 



III 


1 


A 


1^4 








m 




Wm 






328 



LOGARITHMS. 



TT. — Tf llio nogativo clmractcristio is nni exactly 
rlivisiblc, juJd wluit will iimko it so, both to it and to the 
decimal part of tho logarithiu. Theu proceed with tlu- 
division. 

Example.— Divide tho logarithm —4' 887564 by 5. 
4 w antH 1 of bciug divi«iblo by 5; then — 4•8375C4-^5=a 
— 5-f.l-8375G4-i-5=13G7513, tho required logarithm. * 

Rraaon of I. — Tho quotient multiplied by tho divisor must 
give tlio dividend; but [61] a ncgativo quotient multiplied by 
a positive divisor will give a negative dividend. 

Kkahon of II. — In cxanii>le 2, avc luivo merely added -f- 1 
and — 1 to tho same quantity — wliich, of course, docs nC- 
.alter it. 



QUESTIONS. 

1 . What are logarithms ? [32", . 

2. How do they facilitate calculation .? [33] . 

3. Why ia a table of logarithms necessary } [34]. 

4. What is the characteristic of a logarithm ; ant/ 
how is it found ? [37] . 

5. IIoAv ih the logarithm of a number found#by tho 
table.? [38]. 

f). How are the " diflferenccs," given in the tablo 
used.? [30]. 

7. What is the use of "proportional parts .?" [39]. 
S. How is the logarithm of a fraction found ? [41]. 

9. How do we find the logaritlim of a mixed num- 
ber ? [42] . 

10. How is tho number corresponding to a given 
logarithm found ? [43] . 

11. How is a number found when its corresponding 
logarithm is not in the table ? [44]. 

J2. How are multiplication, division, involution and 
evolution effected, by means of logarithms r [46, 48, 
50, and 52]. 

13. When negative characteristics are added, what 
is tlie sign of their sum ? [47]. 

14. What is tlio process for division, when tho cha- 
racieristic of the divisor is negative ? [49] . 

15. How is ancgTitivc characteristic umltiplied r [51]. 
10. How is a negative charactori.stio divided .? [53] 



5 



5 



329 



SECTION X. 



PROGRESSION, &o. 

1. A profTreasion consi.sts of <*i nnmbnr of quantities 
Jill cm a sing, or decreasing l)y fi ciirtaiu law, and forming 
^vliat uro called cmitmued propor/Umnts. When the 
terms of the series coui*tantly increase, it is said to 
l>«) an ascending^ but when they decrease (increase to 
(he /rft), a descending scries. 

2. in an fqnidiO'crcnt or a ?-i/Awt'/(!"mZ progression, tho 
qnantities inci-ease, or decrease by a annvion difference. 
Tiius 5, 7, 9, 11, &.O., is an ascending, and 15, 12, 9, 6, 
&c., ia a descending arithmdical series or progrtission. 
The common diflerencc in the former is 2, and ni the 
latter '^. A continued proportion may bo formed out 
T)f such a series. Thus — 

5 : 7 : : 7 : 9 : : 9 : 11, &c. ; and 15 : 12 : : 12 : 9 : : 
9 : 0, &o. Or we may say 5 : 7 : : 9 : 11 : : &c. ) and 
15 : 12 :: 9 : 6 :: &c. 

3. In a geometrical or equirallonal progression, tho 
quantities increase by a common ratio or multiplier. 
Thus 5, 10, 20, 40, &c. ; -and 10000, 1000, 100, 10, &c., 
are geometrical series. The common ratio in the former 
case is 2, and the quantities increase to the right ; in 
the latter it is 10, and tho quantities increase to tho 
left. A continued proportion may be formed out of 
Huch a series. Thus — 

5 : 10 : : 10 : 20 : : 20 : 40, &c. ; and 10000 : 1000 : : 
1000 : 100 : : 100 : 10, &c. Or wo may say 5 : 10 : : 20 : 
40 : : &c. ; and 10000 : 1000 : : 100 : 10 : : &o. 

4. The first and last terras of a progression are called 
its extremes, and all the intermediate terms its means. 

5. AritliMeiiad rrogression.—To find the sum of a 
series of terms in arithmetical progression — 

j^^yij,;, — Multiply the sum of the extremes uj nan 
the number of terms. 



I 



330 



PROGRESSION. 



_ Example.— Whafe is the sum of a series of 10 terms, tlio 
tirat being 2, and last 20 '? Ans. 2-f 20x •2-'=110. 

C. Reasoiv of the RuLic.— This rule can be easily proved. 
For tins purpose, set down the progression twice over— but 
in sucli a way as that the last term of one shall be under the 
nrst term of the other series. 

Then, 24+21+184-154-12-f 9=the sum. 

9-f 12-f-15 4-18-i-21+24=the aum. And, 

adding the equals, 3;J+33-j-334-33+33+33=twice the sum. 
That is, tf^ir.e tlie sum of the series will be equal to the sum 
of aa ii.any quantities as tliere are terms in the series— each 
of the quantities being equal to tlie sura of the extremes. 
And the sum of the series itself will be equal to half as much, 
or to the sum of the extremes taken ha/f as many times as 
tliere are terms in flie series. The rule might be proved 
correct by any other example, and, therefore, is general. 

EXERCISES. 

1. One extreme is 3, the other 1.5, and the number 
of terius is 7. What is the sum of the series ? Ans. 63. 

2. One extreme is 5, the other 93, and the number 
of terms is 41). ^VHiat is the sum .? Ans. 2401. 

3. One extreme is 147, the other |, and the number* 
of terms is 97. What is tlie sum ? Ans. 7165-875. 

4. One extreme is 4^-, the other 143, and the num 
ber of terms is 42. What is the sum > Ans. 3094-875 

7. Given the extremes, and number of terms — to fini 
the common difference — 

lluLE. — Find the difference between the given ex- 
tremes, and divide it by one less than the number of 
terms. The quotient will bo the common difference. 

Example.— In an arithmetical series, the extremes are 21 
and o, and the nunibcr of terms is 7. Wha^; is the common 
diilerence '? 

21 — 3-^7 — 1 = 18-i-6 = 3, the required number. 
8. Reasov of thk Rule.— The diflference between the 
greater and lesser extretne arises from the common ditferenco. 
bemg aiJded to the lesser extreme once for every term, ex- 
cept tbe lowest ; that is, the greater contains the lesser extreme 
plus the common difference taken once less than the number 
of terms. Therefore, if we subrract the lesser from the greater 
extrcriiu, the diifereuce oblaiuod will be equal to the common 
dil}vM-en'»e multiplied by one less than the number of terms 
And if wo divide tlio difference by one less than the number 
of tcrm.^ we will have the cuininun difference. 



5. 



Tb 
and 497, 
common 

6. Th 
and 9|, : 
common 

7. Th 
and I, a: 
common 

9. To 
two give 

Rule 
cording 
it to, or 
term ; a 
the thirc 
ing tern 

Wen 
terms is 

EXAMI 

21. 21- 

the seri( 
6 . 



Exam 
10. 30 

the Borl 



8. r 

A:n3. 4 
9. 1 

Ans. 1 
10 

Am. 6 



PnOftRESSION. 



331 



EXERCISES. 

5. The extremes of an arithmeti(;al series are 21 
and 497, and the number of terras is 41. What is the 
common difference .? Aiis. 11-9. ^ m-ro, 

6 The extremes of an arithmetical serieR are 127«a 
and 91, and the. number of terms is 26. What is the 
common difrerenee ? Aiis. 4^. ^ 

7 The extremes of an arithmetical series are 77|f 
and*!, and the number of terms is 84. What is the 
common difference ? A^is. |f . 

9. To find a7ii/ number of arithmetical means between 

two given numbers — 

Rule.— Find the common difference [7] ; and, ac- 
cordinc' as it is an ascending or a descending series, add 
it to, or subtract it from the first, to form the f oond 
term ; add it to, or subtract it from the second, to torni 
the third. Proceed in the same way with the remain- 
ing terms. , , i. <? 

VVe must remember that one less than the number ot 
terms is one more than the number of means. 

Example l.~Find 4 arithmetical means between 6 and 

21. 21-6 = 15. TTT==''^) th^' common difference. And 

the series is — „ « . r o 

6.6+3. 6+2x3 . 6+3x3 . 6+4x3 . 6+5x3. 
Or 6 . 9 . 12 . 15 . 18 . 21. 

Example 2.— Find 4 arithmetical means between 30 and 

10 30-10=20. TITT^^' *^^ common difference. And 
4+i 

the Bcrics ia — io ^ i i in 

30 . 26 . 22 . 18 . 14 . 10 

This rule is eyideut. 



EXERCISES. 

8 Find 11 arithmetical means between 2 and 26 
A:m. 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24. 

9 Find 7 arithmetical means Detween 8 and 64 
Ans. 11, 14, 17,20, 23,26, 29. 

10 Find b arithractieal means between 4J-, and id^ 
Am. 6,7J,9, 101, 12. 



332 



rUOGRESSION. 



10. Given the extremes, and tlie number of terms — 
to find any term of an arithmetical progression — 

Rule. — Find the common difference by the last rule, 
and if it is an ascending series, the required term will 
be the lessor extreme plus — if a descending series, the 
greater extreme minus the common difference multiplied 
by one less than the number of the term. 

Example 1. — What is the 5th term of a series containing 
9 terms, the first being 4, and the last 28 ? 

28-4 ^ 

— Q — =3, is the common difference. And 4-f 3x5 — 1= 

16, is the required term. 

Example 2. — What is the 7th term of a series of 10 terms, 
the extremes being 20 and 2 ? 
20-2 ^ _ „ 

— n— =2, is the common difference. 20 — 2x7—1=8. 

is the required term. 

11. Reason OF the Rxtle. — In an ascending series th» 
required term is greater than the given lesser extreme to the 
amount of all the differences found in it. But the number of 
differences it contains is equal only to the number of lerir.a 
which precede it — since the common difference is not found in 
the^trs^ term. 

In a descending series the required term is less than the 
given greater extreme, to the amount of the differences sub- 
tracted from the greater extreme — but one has been subtracted 
from it, for each of the terms which precede the required term. 

exerclses. 

11. In an arithmetical progression the extremes are 
14 and 86. and the number of terms is 19. What is 
the 11th term .? Ans. 54. 

12. In an arithmetical series the extremes are 23 and 
4j and the number of terms is 7. What is the 4th 
ttrm } Ans. 13. 

13. In an arithmetical series 49 and £ are the ex- 
tremes, and 106 is the number of terms. What is the 
94th term } A^is. 6-2643. 

12. Given the extremes, and common difference— to 
find the number of terms — 

11 ule.— 'Divide i\\Q differcnco bctAvccn the given ex- 
tremes by the common difference, and the quotient plus 
unity will be the number of terms. 



Ex A Mr 1 
which th 
ence 3 '? 
20- 



3 



13. He, 
lesser ext 
terms, i 
except th 
the extre 
■will be C3 



14. Ii 

and 12, 
number 

15. Ii 
and 32, 
number 

16. I 



13 



'<.}•) 



an I 



number 

14. G 
and one 

llULE 

terms, n 
The difi 

EXAMI 

the nun 
AVhat is 



15. Rj 

Slim = 8 

divide ec 

vre shall 

2 X 

the num 

tractiug 
have 



rKOGREHSION. 



833 



ExAMiM.K. — How many tenns in jin firUhinotical series of 
which tho extromea arc 5 and 20, and Iho common ditfer 
eiicc 3 ? 



20-5 



3 



=7. And 74-1=8, is tlie number of terms. 



13. llEAscN OF THE Rui.E.— The srcatcr diifcrs from tho 
lesser extreme to tlie amount of the diircrcncos found in all the 
terms. But th» common difference is found in all the terms 
except the lesser extreme. Therefore the difference between 
the extremes contains the common difference once less than 
will be expressed by the number of terms. 

EXERCISES. 

14. In an arithmetical series, tlie extremes arc 96 
and 12, and tlie common difference is 6. What is tha 
number of terms ? Ans. 15. 

15. In an arithmetical series, the extremes are 14 
and 32, and the common difference is 3. What is the 
number of terms } Ans. 7. 

16. In an arithmetical series, the common difference 
is A, and the extremes are 14f and il. What is the 
number of terms ? Ans. 8. 

14. Given the sum of the series, the number of terms, 
and one extreme — to find the other — 

lIui.E. — Divide twice the sum by the number of 
terms, and take the given extreme from the quotient 
The difference will be the required extreme. 

Example.— One extreme of an arithmetical scries is 10 
the number of terms is 6, and the sum of the series is 42 
What is the other extreme 1 



2X42 







-10 = 4, is the required extreme. 



15. Reason of the Rule.— We have seen [5] that 2 X the 
sum = sum of the extremes X the number of terms. But if wo 
divide each of these equal quantities by the number of terms, 
■we shall have , o. 

2 X the sum sura of extremes X the number of terms 



the number of terras 
2 X the sum 



the number of terms 

, — „ =s sum of the extreme!?. .And sub- 

the number oi terms 
tractiug the same extreme from each of these equals, wo shall 

have 



ur 



H34 



i>K()(;kxs«u)!v. 



„ X ''"^J^"J^ — onocxtioaie=the sum of the extremes 

the number of tonus 
tlie aaiue extreme. 

twice the sum 
Or ;^i.: „i rrr.:F7;-~:z minus one extreme = the other ox- 



tho number of terms 



a-erae. 



EXERCISES. 



17. One extreme is 4, the number of terms is 17, 
/md the sum of the series is 884. What is the other 
extrcuio ? Ans. 100. 

18. One extreme is 3, the number of terms is 63, 
and the sum of the series is 252. What is the other 
extreme ? A71S. 5. 

19. One extreme is 27, the number of terms is 26, 
and tlie sum of the scries is 1924. What is the other 
extreme.? Ans. 121. 



16.^ Geometrical Progression. — Given the extremes 
and common ratio — to find the sum of the series — 

Rule. — Subtract the lesser extreme from the product 
of the greater and the common ratio ; and divide the 
difference by one kss than the common ratio. 

Example. — In a geometrical progression, 4 and 312 are 
the extremes, and 'he common ratio is 2. What is the sum 
of the series. 

312x2 — 4 

— 2 — Y — ~ ^= ^^^' *^® required number. 

17. Reason of the Rui^e. — The rule may be proved by 
setting down the series, and placing over it (but in a reverse 
order) the product of each of the terms and the common ratio. 
Then 

Sum X common ratfo = 8 4- 16 + 32, &c. . + 81 2 -f- 624 
Sum= 4 + 8 + 16+32, &c. . 4-312 . 

And, subtracting the lower from the upper lino, we shall have 
Sum X common ratio — Sum = 624 — 4. Or 

Common ratio — 1 X Sum = 624 — 4. 

And, dividing each of the equal quantities by the common 
ratio minus 1 

642 (last term X comrrfon ratio") — 4 (the firgt term) 

Sun = ^^- ^ vv— — 5, — ^^ ~ 

common ratio — 1 

Which ib the rule. 



20. The 
2, and th 
Ans. 682. 

21. The 
175092, an 
Ans. 1932 

22. The 
ore yV anc 
the sum.? 

Since the 

23. The 
937-5, and 
A.)is. 1171 

18. Giv 
goometricii 

RULE.- 

by thc! les! 
i.s indicate 
bo the req 

EXAMPI-F 

pro<:;rcs8U)v 
11 1011 ratio 'i 
SO 

19. Tiv.^u 

to the Icsse 
tlie comuioi' 
pinuo the c> 
is, tlie gren 
power imlic 
tiplio'l by t 
by the I0S.S1 
is indicate; 
obtain the 



24. Th 

common 1 
85. Til 



I'R0(iRE.S3I0N 



335 



EXERCISKS. 

20. The extremes of a geometrical pciies are 512 and 
2, and the common ratio is 4. What is the sum ? 
Ans. 682. 

21. The extremes of a geometrical series are 12 and 
175092, and the common ratio is 1 1. What Is the sum ? 
Ana. 193200. 

22. The extremes of an infinite geometrical series 
ore yV and 0, and j\ is tlie common ratio. What is 
the sum.? Ans. -i. [Sec. IV. 74.] 

yinco the sorics is infinite, the lessor extreme=0. 

23. The extren\es of a geometrical series are *3 and 
937-5, and the coiumou ratio is 5. What is the sum ? 
A)is. 1171-875. 

18. Given the extremes, and number of terms in a 
geometrical series — to find the common ratio — 

EuLE. — Divide the greater of the given extremes 
by the lesser ; and take that root of the quotient which 
i.s indicated by the number of terms minuni 1. This will 
be the required number. 

ExAMPi-E.— 5 and 80 arc the extremes of a geometrieal 
progression, in which there arc 5 terms. ^Vhat i^ the eoiu- 
11 ion ratio ? 

7-=lG. And ^'1G=2, the required common ratio. 
o 

19. REASON OT THK Rule. — The greater extvemc is o.i|nnl 
to the lesser multiplied by a product whicli has for its factors 
tlie conmioii ratio tukeri once loss tiian the number of terms — 
since the comni<in ratio is not found in tiie Jiist term. That 
is, tlie trreater cxtre\ae contains the common ratio raised to a 
power indicated by 1 less than the number of terms, and mul- 
tiplied by the lessor extreme. Consequently if, after dividinj; 
by the lessor extreme, we take that root of the quotient, which 
is indicated by one less tlian the number of terms, we sholl 
obtain the common ratio itself. 



EXERCISES 



24. The extremes of a geometrical series are 4911^2 
oj^fl 3^ atul the number of terms is 8. What is th3 
common ratio } Ans. 4. 

2d. The extreuies of a geometrical series arc 1 and 



336 



PROGRKSSION, 



15625, and the number of terms is 7 What is the 
common ratio ? Ans. 5. 

26. The extremes of a geometrical series arc 
20176S035 and 5, and the number of term* is 10 
What is the common ratio ? Ans. 7. 

20. To find any number of geometrical means be 
two on two quantities — 

Rule. — Find the common ratio (by the last rule)^ 
and — accovdina; as the series is ascendinoj, or descend- 
ing — multiply or divide it into the first term to obtain 
the second ; multiply or divide it into the second ta 
obtain the third ; and so on with the remaining terms. 

We must remember that one less than the number 
of terms is one more than the number of means. 

Example 1. — Find 3 geometrical means between 1 and 
81., 

^--=:3, the common ratio. And 3, 9, 27, are the re- 
quired means. 

E-XAiMPLE 2. — Find 3 geometrical means between 12r)f 
und 2. 

1250 _ . . 1250 1250 1250 



4/-2-=''>- And 



H, or 250, 50, ]';' 



5 5x5 5x5x5 

are the required means. 

This rule requires no explanation. 
EXERCISES. 

27. Find 7 geometrical means between Sand 19683 ' 
Ans. 9, 27, 81, 243, 729, 2187, 6561. 

28. Find S geometrical means between 4096 and Si" 
Ans. 2048, 1024, 512, 256, 128, 64, 32, and 16. 

29. Find 7 geometrical means between 14 and 



23514624.? Ans. 84, 504, 3024, 



18144, 108864, 



653184, and 3919104. 

21 . Given the first and last term, and the number of 
terms — to find any term of a geometrical series — 

lluLE. — If it bo an ascendhig series, multiply, if a 

df'sp.or>diTi'» sorins). divJ;<ft tho firs.t, tomi hv> fl 'ij imwrrr 

of the common ratio which is indicated by t;*; num&ei 
of the term minus 1. 



is the 



30. 



32. 



PROGKESSION. 



337 



Example 1.— Find tlio 3nl term of a {reometrical scries, 

of %vhich the tiryt term is (3, the last 1458, suid the number 

ol" term.-) G. 

1458 
The common ratio is ;j/-^-=P). Tlicveforc the required 

term is 0x3^=54. 

]'>xAMPi,K 2. — Find the 5th term of a series, of which tlw 
extremes are 524288 and 2, and the nundjer of terms is 10. 

5242SH . 5242NS 

Iho common ratio ^ — rr~— 4. And — j4- = 2048 

is the required term. 

22. Rkasoiv of the Riti.k.— Tn an ascending series, any 
term is the proihict of the lirsi and the couinion ratio taken 
as a factor so many times as there arc preceding terms — siiKsa 
it is not found i7i ihe fust term. 

In ade-ccndiug series, nny term is eqnal to the first term, 
divided by a product containing the common ratio as a factor 
so many times as tliero arc pi-eeoding terms — since evei-y term 
but that whieh is required adds it once to the factors wducli 
coDstitute the divisor. 

EXERCISES. 

30. Wh;it is tho 6th term of a series having 3 and 
5851)375 as extremes, and, containing 10 terms .^ Ans. 
9375. 

31. Given 39366 and 2 as tho extremes of a series 
Laving 10 terms. What is the 8th term.? Ans. 18. 

32. Given 1959552 and 7 as tlio extremes of a series 
havinff 8 terms. What is the Gth term ? Ans. 252. 

O 

23. Given the extremes and common ratio — to find 
the number of terms — 

Rule. — Divide the greater by the lesser extreme, 
and one more than the number expressing what power 
of common ratio is equal to the quotient, will be tho 
re(|ulrcd qiumtity. 

ExAMi'Lic. — How many terms in a series of which tho 
extriMnes are 2 and 25G, and the conuuou ratio is 2 ? 

-,-^-=128. But 2"=::128. There are, therefore, 8 term£3. 

The common ratio is fonnd as a factor (in the quotient of 
tho greater divided by tho lo.'^ser extreme) once less than the 
number of terms. 



II L 



838 



PROGRESSION. 



KXERCISES. 

33. TIow many terms in a xories of wliich the first, is 
78732 and the last 12, and the cummun ratio is 9 .-* 
Ann. 5. 

3-4. IIoAV niany terms in a series of wliieli llic ex- 
tremes and common ratio are 4, 47()r)9f;, and 7 ? Avs. 7. 

35. How inany terms in a series of Avliieh the ex- 
tremes and common ratio, are 19GGUb!, G, and 8 f Ans. 6. 

24. Oiv^en the common ratio, number of terms, and 
one extreme — to find the other — 

lluLE. — If the lesser extreme is given, multiply, if 
the greater, divide it by the common ratio raised to a 
power indicated by one less than the number of terms. 

ExajMpi.k 1. — In a g;oomctrioal series, the lesser extreme 
is 8, the number of terms is 5. and the enniinoii ratio is Gj 
what is the other extreme '? Atis. 8xG'~'=10oG8. 

Example 2. — In a goomefrioal series, the greater extreme 
is 0561, the number of terms is 7, and the common ratio is 
S; what is the other extreme'? Jnx. G5Gl-j-3'~'=U. 

This rule does not require any exphvnatiou. 

EXERCISES. 

36. The common ration is 3, tlio number of terms is 7, 
and one extreme is 9 ; what is the other ? Ans. 0561. 

37. The common ratio is 4- the number of terms is 
6, and one extreme is 1000 ; what is the other ? Ans. 
1024000. 

38. The common ratio is 8, the number of terms i** 
10, and one extreme is 402653184 ; what is the other ? 
Ans. 3. 

In progression, as in many othe- rules, the application of 
algebra to the reasoning '.voukl greatly simplify it. 

MISCELLANEOUS EXERCISES IN PROGRESSION. 

1. The clocks in Venice, and some other places stiilvO 
the 24 hours, not beginning again, as ours do, after 12. 
How many strokes do tliey give in a day ? Ans. 300. 

2 A butcher bought J 00 sheep; for the first ho 
gav(' 16". , and for the larit i.'9 lO.:;. What did ho pny for 



Of 

Wl 



all, suppi 
Ans. £d 

3. A 
yard he 
price of ; 

4. A] 
the first 
on, until 
did he tr 

5. A 
that the 
and that 
year. I 

6. Fii 
Ans. ' 

7. 

8. 

payment 
being £ 
common 
the ratio 

9. Wl 
thing fo: 
second, 
shoe .'' 

10. A 
queathec 
gave £1 
next, li 
was the 
of the I 
ceived i 



1. W 

series ? 

2. W 
trical pi 
names .^ 

3. W 
ratio ? I 



PROGRESSION. 



339 



nil, supposiig tlicir prices to form an arithmetical scries ? 
Ans. iioOO. 

8. A person bought 17 yards of cloth ; for the first 
yard he gave 2.?., and for the last IOj. What was the 
price of all ? Ans. £r> 2s. 

4. A person travelling into the country went 3 miles 
the first day, 8 miles the second, 13 the third, and so 
on, until ho went 58 miles in one day. How many daya 
did he travel? Aois. 12. 

5. A man being asked how many sons he had, said 
that the youngest was 4 years old, and the eldest 32, 
and that he had added one to his ftimily every fourth 
year. How many had he ? Ans. 8. 

6. Find the sum of an infinite series, -J-, ^j -gVj &c. 
Ans. 1. 

7. Of what value is the decimal -463' ? Ans. ^f f . 

8. What debt can be discharged in a year by montlily 
payments in geometrical progression, the first term 
})eing jei, and the last £2048; and what will be tho 
common ratio ? Ans. The debt will be £4095 ; and 
the ratio 2. 

9. What will be the price of a horse sold for 1 far- 
thing for the first nail in his shoes, 2 farthings for the 
second, 4 for the third, &c., allowing 8 nails in each 
shoe ? Ans. £4473924 55. 3frZ. 

10. A nobleman dying left 11 sons, to whom he be- 
queathed his property m follows ; to the youngest he 
gave £1024; to the next, as much and a half; to tho 
iiext, 11 of the preceding son's share ; and so on. What 
was the eldest sou's fortune ; and what was the amount 
of the nobleman's property r Ans. The eldest son re- 
ceived £59049, and the father was worth £175099. 

QUESTIONS. 

1. Wliat is meant by ascending and descending 
series ? [1]. 

2. What is meant by an arithmetical and geome- 
trical progression ; and are they designated by any other 
names ? [2 and 3] . 

3. What are the common difierence and common 

ratio } [2 and 3] . 



i 




II 



540 



ANNf ITIKS 



4 t\).\>^ that a onntiimofl proportion mny 1)0 fomied 
froTi, a 8crics of ehhov kiiul r [2 iind 3J. 

.'j. Wliat arc moans ii"/i (^xtrcniej ? [4]. 

6', l^ow is the hm, aritniuetical or a gcomo* 

trical series fouiivl ? jo auu Itij. 

7. [low is the common difference or common ratio 
found ? [ 7 and 1 8] . 

8. How is any nimiber of aritliraetioal or geometrical 
means fouwd ? [9 and 20] . 

9. How is «ny particular arithmclicul or geometrical 
mean found ? [10 and 21]. 

10. How is tlio number of terms in an arithmetical 
or geoinctriual series found .? [12 and 23]. 

11. How is one i^xtremo of an arithnuitical or geome- 
trical scries found } [14 and 24]. 



ANNUITIES. 

25. An annuity is an income to he paid at stated 
times, yearly, half-ye;irly, &o. It is either in possession ^ 
that is, entered upon alread}'', or to he entered upon 
immediately ; or it is in reversion^ that is, not to com- 
mence until after some period, or after something has 
occurred. An annuity is certain when its commence- 
ment and termiuation are assigned to definite periods, 
conlingcMt when its l)Oginning, or end, or both are 
uncertain ; is in arrears when one, or more payments 
are retained after they have become due. The amount 
of an annuity is the t.im of the payments forborne (in 
arrears), and the interest duo upon them. 

When an annuity is paid off at once, the price given 
for it is called its ]n-escnl: worthy or value — which ought 
to be such as would- if left at compound interest until 
the annuity ceases — 'produce a sum fqual to what would 
be due from the annuity left unpaid until that time. 
This value is said to l;»^e so many years'* purchase ; that 
is, so many annual payments of the income as would "be 
just equivalent to it. 

26. To find the amount of a certain number of pay- 
ments in arrears, and the interest due on them — 



iJlJT.F..— 

the sum of 
bo the vci[\ 

EXAMPI.K 

uuiialdfor C 
The Inst, 
them, form 

4 . . XlX^^ 

Example 
is unpaid 1 
por emit, p 

In this c 

with its in 

luulli plied 

with its ill 

Ik the %\\i 

''i'lie unu 

tlio trouti^■ 

the series 

X2-100U2) 

• ~1'( 

The am 

same aa t 
years, whi 
sov, is eq 
Xl; that 

the roquii 

It vou 

ar.nuiiV ! 
2-527 - 

to bo CO 1 

Hence 
payment 
them — 

Subtr 
number 
interest 
tirnt by 



ANNUITIES 



liUT.F.. 



341 

llion 
aud interest duo on tlicm, will 



-Find the interest dtie on each payment 

payinoi 
anioun 



uin, 



Iho smii of the p 
Ic tho reiiUKtrd 

Faampi.k 1.— Wliat will he the amount of XI .per ann 
ui mafd for G years, 5 per cent, simple interest heing allms .dT 

The last, and , ^-oeedingpaymeni.^vith the inteij^s^^ 
theui, form tho .'. ^tfnnetical sov.e. ^'-^.^''^^^^^-^ 
4 XlxXOS XI. Audits sura 18 X14-X1+-^^'>X'X 
|=X24-X'25X3=XG'75=X0 15.s., the required amount. 

ExAMi'LK 2.-If the rent of a {\irm worth XGO r;;^ a.nnuni 
is unpaid fo^ 10 year., how much doen u amount to, at . 
pt>r cent, per an. compound mtercst . 

In thi« ease the series is g.o,..fr/ra/ ; and the last my^^en 
wilh its interest is the onunmt of XI for IH (IJ — i) J**''^" 
mvlli hod lY tho siven annuity, the pi;eeoding paynie 
with itH iutorvst is Iheamomit of Xi fnr 17 years mulUphed 
hy the ^^iven annuiiy, &:c. ^^^^ ^^^^^ ^,f 

t.^';vc:rTf^l^yiar:^:e2.^^^^^ Then tl. sum of 
the series is — 
££i2!^^X^:2i^55ll'^l?[l()]=:l •i2-4, the required amount. 

The iMivmnt of XI f.a- 18 years mvdtii.lied hy 105 is the 

same as the Huu.imt ol XI toi U, oi ilo ol'' , the divi- 
vo'v^ which is found to he X252/ . And lOo —- 1, trie uuj 
^;^is e ual to the amount of Cl for oue payment imnua 
11'; thltis,tothe Intcre t of XL fc.; one payment. Hence 

X2-52( X <^3i^' _ £18o24. 
the required sum will he - " .q^ 

It would evidently ho th. same ^hin^ to consider^ the 
ar.nui;y as XI, and then multiply the re;-. It hy 00. lluu 

'■^l-Xni X G0 = X18"'2-4. For an annuity of XGO ought 
to ho GO times as productive as one of only XL 
• Hence, briefly, 1 find tlie amount of any nun.her of 
ptiynients in arrears, and the compoum intore.t dae on 

^^'sl^aet £1 from the amount of ^1 for the giverj 
T,mph<u- of pavments, and divide the diilcrenco^by 4Jiq 
interest of £1 for one payment ; then um.lU^^)iy ine 'i">^.^ 
tient l>y the "fiyeii nni. 



I 



^1 



t. 



w 



' ■ ANN urn KS. 

re?J*«?ij''.?H'' "*' 'V.1^I*^^^^«— E'icl» payment, with its Into- 
mn ; K I ^ co„s unto u H.juirnt, a.uu.mt ; „u I tlu, Hu.n dtte 
•» u<t bo the sum of tlie.vo aruounlM-wliiel, Vunu a ,/.«r Wn^ 
•ones, because of tlio deoreash.g interest, ari" ng fro m thf 
.lecreiw.ug number of times of pa^monl, ^ ® 

which 1 nfn « L " °^ ? 7'Mm./,ra/ series, one extreme of 
Which IS tiiehrst payment plus tlio interest due upon it at th« 

aitterence the interest on one payment due at the next. 

Hut when cowpounil interoat is allowed, what is due will hn 
the sum <.f a geoniHrical series, one extreme of wEx^sth« 
hrst payment plus the interest duo on it at tho a^t the otlor 
Jnr/hf •*f^"'?i'/"^* it« common ratio £ pi i'ts tter hI 

inteieat due on the hrat payment at the time of the last will 
bo the intoros due for one less than tho numLr of payments 

payment''"' '' "'' '"^ "° "'° '^«' "^^^^ '''' '^^'^ KoS 

EXERCISES. 

fn.^i^''''^' '" *^, '^^^'^'^"^^ «f ^'37 per unnu.u unpaid 

Ls ^llTios ^'' """*• ^'' """• '''"^^' ^°^"^'^'^- 

2. What is tho amount of m annuity of i^loo to 
continue 5 yeai;s at 6 per cent, per an. conipouua inte- 
rest .'^ Ans.£bmUs.2\d. i- i i^ 

3. What is the amount of an annuity of ^£350 to 

4 What is the amount of ,£49 per annum unpaid 
Ans ^ilu 5 ^^^ ''x''^' ^^^'P^'^"^ ^"*^^est b^^i"g 'lUovved .? 

28. To find the present value of an annuity— 
Rule.— Find (by the last rule) the amount of tho 
given annuity if not paid up to the time it will cease. 
1 hen ascertain how often this sum contains the amount 
ot J^l up to the same time, at the interest allowed. 

Example.— What is the present worth of an annuity of 
S^^^uf^'itm: 18 ye,.s would amount t, 






ANNIHTCKH. 



313 






But. .CI put to intorost for 18 years at the same ruto 
would fimiaiut to X2-40GG12. Tliorolm-o 

— *''40(j(?>~" ~ ^® required value. 

The sura to be puiil for the annuity should evidently bo euch 
ns would produce the aiimo as the annuity itself, ia the same 
tima. 

liXEKCISES. 

f). What is the present worth of an annuity of £27, 
to be paid for 13 years, o per cent. coinpoTind interest 
being allowed ? Ans. £2j3 \2s. (j\d. 

6. What is the present worth of an annuity of J6324, 
to bo paid for 12 years, 5 per cent, compound interest 
beiug allowed? Aiis. £2671 I3s. 10}f/. 

7. What is the present worth of an annuity of .£22, 
to be paid for 21 years, 4 per cent, compound interest 
being allowed } Ans. £308 12s. lOd, 

29. To lind the present value, when the annuity is 
in perpetuity — 

lluLK. — l)ivide the interest which £1 would produce 
in perpetuity into £1, and the quotient will be tno sum 
reijuirud to produce an annuity of £1 per annum in 
perpetuity. iMultiply the (|uutioiit by the number of 
pounds in the given annuity, and the product AtII be 
the required present worth. 

Example. — Wliat is the value of an income of £17 for ever * 

Let us suppose that XlOO would produce £b per cent, per 

an. for ever: — thou £i would produce £-05. Therefore, 

to produce £1, we roquire as many pounds as will be equal 

to the number of times £-05 i,<* contained in £1. But-7r?=ai 

£20, therefore £20 Avould produce an annuity of £1 for 
ever. And 17 times as much, or £20x17=340, which 
would produce an annuity of £17 for over, is the required 
present value. 

EXERCISES. 

8. A small estate brings £25 per annum ; wh&t is 
its present worth, allowing 4 per cent, per .annum irste- 
rest .? Ans. £62.5. 

9. What is the present worth of an incorae of £347 






; 1 



I 



344 



ANNUITIES. 



in perpetuity, allowing G per cent, interest? Ans 
£5783 6s. S(L 

10. What is the value of a perpetual annuity of £46, 
aUowiug 5 per cent, interest ? Arts. ^£920. 

_ 30. To find the present value of an annuity in rever- 
sion — 

lluLE. — Find the amount of the annuity as if it were 
forborne^ until it should cease. Then fi^d-what sum, 
put to interest now, would at that time produce the 
same amount. 

Example.— What is the value of an annuity of £10 per 
annum, to continue for 6, but not to commenr- for 12 years, 
o per cent, compound interest being allowed ? 

^'^1 ""J^^ofe^ ^^^ ^'^^ ^ ye^^'« if left unpaid, would bo 

l'?'!^-u^-a'^]^} ' ^^'1 ^1 ^0"!^' i» 1^ years, be worth 
i^ii'ObUoO. Ihercforo 

M8-0VJI 

ll~08959~^"^ ^^' '^'■^•■> i^ ^^^^ required present worth. 

EXERCISES 

11. what is the present worth of .£75 per annum, 
which is not to commence for 10 y^ars, but will con- 
tinue 7 years after, at 6 per cent, compound interest ? 
Am. £1.55 9.?. 7-^d. 

12. The reversion of an annuity of £175 per annum, 
to continue 11 years, and commence 9 years hence, is to 
be sold ; what is its present worth, allowing 6 per cent, 
per annu.n compound interest ? Ans. £430 7*. }d. 

13. What is the p^-esent worth of a rent of £45 per 
annum, to commence in 8, and last for 12 years, 6 per 
cent, compound interest, payable half-yearly, being 
allowed.' ^?w. £117 25. S^^. 

31 When the annuity is contingent, its value depend.^ 
on the probability of the contingent circumstance, or 
circumstances. 

A life annuity is equal to its amount multiplied by ' 
the value of an annuity of £1 (found by tables) for tho 
given age. ^ The tables used for the purpose are calcu- 
l:i ted on principles derived from the doctrine of chances, 
observations on the duration of life in different circum- 
stances, the rates of compound interest, &c. 






POSITION. 



345 



QUESTIONS. 

1 . What is fin annuity ? [25'j . 

2. What is au annuity in possession — in reversion — 
certain — contingent — or in arrears ? [25]. 

3. What is meant hy the present worth of an an- 
nuity ? [25] . 

4. Plow is the amount of any number of payments 
in arrears found, the interest allowed being simple or 
■compound ? [26] . 

5. How is the present value of an annuity in posses- 
sion foimd ? [28]. 

6. How is the present value of an annuity in per- 
petuity found ? [29] . 

7. IIow is the present value of an annuity in rever- 
sion found .' [30]. 



^' 



i 1 






POSITION. 

32. Position, called also the " rule of false," is a rule 
which, by the use of one or more assumed, but false 
numbers, enables us to find the true one. By means of 
it we can obtain the answers to certain questions, which 
we could not resolve by the ordinary direct rules. 

When the results arc really proportional to the sup- 
pof'ition — as, for instance, when the number sought is 
to be muUijflied or divided by some pi'oposed number ; 
or is to bo increased or diminished by itseJf, or by some 
given mnUijplc. or j)art of itself — and when the ({uestion 
contains only one p)-opositio7ij we use what is called 
single position, assuming only one number ; and tho 
(juantity found is exactly that which is required. Other- 
wise — as, for instance, when the number sought is to bo 
increased or diminished by some absolute number, which 
is not a knovrn multiple, or part of it — or when two 
propositions, neither of which can be banished, are con- 
tained iu the problem, we use douhk position, assuming 
itjuo numbers. If the number souiclit is, durins; tho 
process indicated by the question, to be involved or 
evolved, we obtain only au approximation to the quan- 
tity required. 



I 



Ill 




346 



POSITION. 



.'«. ^^ingk PosiUon.-~B.vhK. Assume a number, and 
perform with it the operations described in the question ; 
then say, as the result obtained is to the number nse(l\ 
so IS the true or given result to the number required. 

ExAMPLK.— What number is that whicli, being multiplied 
^J -». '^y i, and by 9, the sum of the results shall be 231 ? 

Let us assume 4 as the quantity sought. 4x5-j-4x74- 
4x0=84. And 84 : 4 :: 231 : ^^--ll, the required 
number. 



84. Reasoiv of the Rule.— It is evident that two num- 
bers, mnlUpaed or divided by the same, should produce pro- 
poi t.onatc results -It is otherwise, ho^yever, when tJie ^me 
quantity is added to, or subtracted froia tliem. Thus let tho 
given question be changed into the following. What number 
IS that wiuch being multiplied by 5, by 7, and by 9, the sum 
ol the products, plus 8, shall bo equal to 239 .> 

Assuming 4, the result will be 92. Then we cannot say 
92 (84+8) : 4 : : 239 (231-|-8) : 11. 

For though 84 : 4 : : 231 : 11, it does not follow that 
h4-j-8 : 4 :: 2ol-j-8 : 11. Since, while [Sec. V. 29] we may 
multiply or divide the first and third terms of a geometrical 
proponiun by the same number, we cannot, without destroy- 
ing the proportion, add the same number to, or subtract it 
trom thorn. The question in this latt«r form belongs to tho 
rule of duuble pcsitiou. 

EXERCISES. 

1. A teacher being asked how many pupils he had^ 
replied, if you add l, -] , and J- of the number together, 
the sum will be IS ; what was their number > Ans. 24. 

2. What number is it, which, being increased by \, 
J , and i of itself, shall be 125 } Ans. 60. 

3. A gentlcnuin distributed 78 pence among a num- 
ber of poor persons, consisting of men, women, and chil- 
dren ; to each man he gave GiL^ to each woman, 4rZ., 
and to each child, 2d. ; there were twice as many 
women as men, and three times as many children as 
women. How many were there of each } Ans. 3 men, 
f^ women, and IS children. 

^ 4. A person bought a chaise, horse, and harness, for 
£t){) ; tbe horse came to twice the price of the harness, 
and the chaise to twice the price of the horse and har- 



POSITION. 



347 



flCKS. "What did he give for each ? Am. He gave for 
ihc harness, £C) I'Ss. 4(1. ; for the horse, i^^lS ijs. Sd. ; 
uud for the chaise, iD'lO. 

5. A's age is double that of J5's ; IVs is trchlo that 
of C's ; and the sum of all their ages is 140. What is 
the age of each ? Ans. A's is S4, J3's 42, and C's 14. 

6. After paying away J- of my money, and then } of 
the remainder, I had 72 guineas left. What had I at 
Krst ? yh^s. 120 guineas. 

7. A can do a piece of work in 7 days ; J* can do the 
same in 5 days ; and C in G days. In what time will 
ull of them execute it ? Ans. in Ij^^ days. 

8. A and B can do a piece of work in 10 days ; A 
by himself can do it in 15 days. In what time will 13 
do it ? Ans\ In 30 days. 

9. A cistern has three cocks ; when the first is opened 
all the water runs out in one hour ; when the second is 
opened, it runs out in two hours ; and when tlic tlurd i? 
opened, in three hours. In what time Avill it run out, it' 
ail the cocks are kept open together r Ans. In /y hours. 

10. What is that number whose ■}, J-, and -} parts, 
taken together, make 27 ? Ans. 42. 

11. There are 5 mills; the first grinds 7 bushels of 
corn in 1 hour, the second 5 in the same time, the third 
4, the fourth 3, and the fifth 1. In what time will the 
five grind 500 bushels, if they work togctlier ? A71S. 
In 25 hours. 

12. There is a cistern which can be filled by a cock 
in 12 hours ; it has another cock in the bottom, by 
which it can be emptied in IS hours. In what time will 
it be filled, if both are left open ? Ans. In 36 hours, 

35. Doiibh Position . — Rule I. Assume two con- 
venient numbers, and pe.'form upon them the processes 
supposed by the question, marking the error derived 
from each with + or — , according as it is an error of 
e.iwvs, ©r of defect. Multiply each assumed number into 
the error which belongs to the other ; and, if the errors 
are hot/i plus, or hoth minus, divide the diJJ'crence of the 
products by tho difference, of the errors. ]3ut, if one is 
a plus, and the other is a minus error, divide the sum of 



I M 






348 



POSITION. 



the products by tlio mm of the errors. I,, either cas(^ 
the result will bo the nuiriber 60U'.-lit, or an apuroxi 
iiiatiuu to it. ^ ■ 

Exami'm: 1.— If t„ 4 timpg the price of my horse XIO U 
added, the sum will be £100. AVliat did it cast '? 
As.suming muubers wliich give two errors oi" excess-^ 
First, lot 28 be one of thoni, 
Multiply by 4 

112 
Add 10 

From 122, the result obtained, 
subtract 100, the result required, 

and the remainder, +22, is an error of crrew. 

iMultiply by 31, the other assumsd uuinlwr 

and 082 will bo the product. 
Next,, let the assumed number be 31 
Multiply by i 

124 

Add 10 

From 134, the result obtained, 
subtract 100, the result required, 

, and the remainder, -{-34, is an error o?nrrc'^<i. 

Multiply by 28, the other assumed uum, 

and 'J52 will bo tlic produot. 
From this subtract G82, thoproductfouud.diove, 

divide by 12)270 
and the required quantity br22r)=r£22 10s. 



Difference of crrora=34-.':;^12, the immber h\ whieli 



._. .lift iiiimhdv ]\x «']ii,.1 

we have divided. 

_ 36. Rkason of thk ?vUle.— AVhen in cxrimnle 1, we mul- 
tiply 28 and 81 by 4, we-iuuitiply the error belongini to e.icli 
by 4. Hence 122 and 134 are, reHpecLivelj, cquuf to tlie true 
result, plus 4 times one of the error.s. Subti-;ictiu<r tO(3, tb<j 
true result, from e;icli of them, we obtahi 22 (4 tiiue.ri >e errur 
in 28) and iJ4 (4 times tlie errur in 81). 

But, as numbers are propurtioniil to their '^y?//miltiples 
the error in 28 : the error in 31 : : 22 (a multiple of the for- 
mer) : 84 (au enuinuiltiple of the latter). 

And from the nature of proportion [Sec. V. 21]— 



POSITION. 



349 



The error in^28x34==thejBiwrM^^ 
Cut 682= tiio~error in"31-f t)ie required number X22. 
And 95'2=the error in 28-|-tiie required uuiabevX34. 
Or, since to multiply quantities under the vinculum [Sco 
[I. 84], we are to multiply each of them— ■ 
t)82=22 times the error in 3i-f-22 times the required number. 
it52=34 times the error in284-3i times the vequiied number. 
Subtracting the upper from the lower line, we shall have 
952—682=34 times the error in 28—22 times the error in 
31-4-84 times the required numbcv— 22 times the required 

number. • oq o i 

But, as -we have seen above, 34 times the error in jy=_'.i 
times the error in 31. Therefore, 34 time:s the error in 28—22 
times the error in 31=0; that is, the two quantities cancel 
each other, and may be omitted. We shall then have 

952 — 082=34 times the required number— 22 times the re- 
quired number; or 270=34-22 (=12) times the required 
number. And, [Sec. V. G] dividing both the equal quanti- 
ties by 12, 

OTA 34—22 

"in- (22 '5) = - T^ *i"^C'* (once) the required number. 

37. ExAiNiPi.E 2.— Using tho same example, and assivming 
nmnbora which ^ixa two errors of defect. 
Let them be 14, and 3.0 — 
14 16 

4 4 



50 
10 

GO, the rosnlfc obtained, 
100, the result required, 

- 34, an error of defect. 
10 



04 
10 

74, tiie result ol)tainod, 
100, the resuk required, 



— 20, an error of defect. 
14 



544 
304 



304 

Difference of errors : 



III 



•I 






: 34 — 20 = 8. 



8)180 

22-5 =£22 lOs., is the required quantity. 

In this example 34=.four times the error (of defect) in 14; 
and 26= four times the error (of detect) in IG. And, yineo 
ftiumbevs are pvoportioptd to iJieir equimultiples, 

The error in 14 : ih - error in IG : : 34 : 2(5. Therefore 
The error in ! lx2G=:tho error i u 16 X34. 
But 544=the required number — the error in 16X81 
And3Gl=thc required number — the error in MX2G 

«i 2 



1 



POSITIOX. 



If wc subtract tho lower from tlic upper lino, we shall Imvo 
644 — o(>4=(rcinuviiig the vinculuia, uud changing tho «igu 
[Sec. 11. It)]) ;M times the requiriMl number— 2(J times tTio 
lequired iiuuiber — ol liiiuis the civui- in 1G4--'J times tho error 
ill 14. 

IJut we found above that 34 times the error in 16=-'26 times 
tlie error in 14. Therefore — 34 times tho error in l<j, nnd4-"<> 
times tlie error in 14=i0, and niiiy be onntted. We will then 
have 544 — ;J(>1=34 times the required number — 2t> times the 
required number; or 180=8 time^ the required number; and, 
dividing botli these equal quantities by 8, 

180 8 

— Q- (22'5) =,-T times (once) tlie requtrcd number. 

38. ExAMi'LE 3. — Using still the same example, and as- 
sinnlng numbers "vvhich will give an error of extaw, and an 
error of defect. 



Let thorn be 15, and 23 -- 

15 

4 

60 
10 

70, tho rosiult ohto.intMl. 
100, tho result required. 

- SO, an error of defect. 
23 



C'JO 
30 



23 
4 

92 
10 

102, tho result obtained. 
100, tho result rcquirod. 

-|-2, an err 01' of exi-e^s. 
15 

30 



Sum of errors = 30 -f- 2 : 



32. 



32)720 

22-5 = £22 lO.s-., the required quantity. 

In this example 30 is 4 times the error (of defect) in 15; 
and 2, 4 times the error (of excess) in 23. And, since numbers 
are proportioned to the equimultiples, 

The error in 23 : the error in 15 : : 2 : 30. Therefore 

The error in 23X30=the error in 15X2. 

But l)UO=::the required iiumber-f-the error in 23x30. 
And 30=the required number — the error in 15X2. 

If Ave add these two linos together, we sliall have b!!0-f 30= 
(removi;ig tlie vinculum) 30 times the required numbor+ 
tvvice the required number -{-30 times the error in 23 — twice 
ihe error in 15. 

But we found above tliat 30 X the error in 23r=2v;;io error 
in 15. Therefore 30!i^the error in 23 --2 X the 'irror in 15=0. 



I'O'MTIO: 



351 



, 



S . w';*'>-> -t.-^- A'ud divldiug each «C tl..e equal 
quantities by 32. 

'^20,<,2.6^=- times (once) the required number. 



8 ' oit 

The given questions might be changed into one belongmg 
to sin^'/e position, thus— 

p„u. times the price f "/.hj^'i =» °X l90.''"wlaf ..id 
or four times the price of "X ''<'''•« '''"j" „„ ^frort of the mind 

39. ExAMPLK 4.-What is that number which is equal to 
4 times its square root +21 '? 



Assume 64 on'' 81- 
4 



n2 
21 



^81= 9 
4 

36 
21 



53, result obtainc'l. 
G4, result ro<iuired. 

81 



57, result obtained 
81, result required 

64 

1536 
891 



13)645 
The first approximation ia 49'6154 

U is evident thatU and 24^ ^Z'lIX^ '^^^ 
„,„„bcrs multiplied or '^"'^'"LteuSe rule is founded, docs 
therefore, as the reason "P°" "™„o"hnation. Substituting 
?,^iV >S;;^Wrfo?r or^-a-ssSeTnuntbers. we ohtatn . 

(Btill nearer approximation. 

TT T7m,l the errors by the last rule ; then 

kind), or their sum ^f they are o^J^"J^ ^ ^^^e of 
error which has been used as multiplier. 






352 



rosiTiox. 



KxAMrr.K. — Taking the Hamo an in tlic lust rule, and a/ 
Burning I'J and 23 as tho reiiuired numbor. 

19 25 

4 4 

70 
10 



8(5 the result obtained. 
100 the result required. 

—14, is error oi defect. 



110 the result o))tainfMl. 
loo the result required. 

-f-10, ia error of excels. 



The errors are of different Y\x\({^; and their sum is 14-f 
ro=24 j and tho difference of the assumed numbers is 25 — 
1U=(}. Therefore 

14 one of the errors, 
is multiplied by 0, by the difference of the numbers. Then 

divide by 24)84 

and 3-5 is the correction for 19, the number 
which gave an error of 14. 

194-(the error being one of defcpt, the correction is to bo 
added) 3 5=22 5=£:i2 IDs. is tho required quantity. 

41. Reason of the Rule. — Tlie diifercnce of tho results 
arising from tlie use of the different assumed numbers (tho 
difference of the errors) : the difference between tho result ob- 
tained b^'' using one of the assumed numbers and tliat obtained 
by using the true number (one of tlie errors) : : the difference 
between the numbers in the former case (the difference betweea 
the a.sHumed numbors) : the difference between thu numbei'S 
. in the latter case (tlie difference between the true mmiber, and 
that arjsumed number wiiich produced tlie erior placed in the 
thir<l term — tliat is the correction required by that assumed 
number). 

It is clear that the difference between the numbers used 
produces a proportional difference in the results. For tlie 
results are different, only because tho difference between the 
assumed numbers has been multipliccl, or divided, or both — 
iu acconlance with tho conditions of the question. Thus, in 
the present iuytsmce, 25 pl'oduoes a greater result thau 19, 
because 0, the difference becween 19 and 25, has been multi- 
plied by 4. For 25x4=s=19x4-f-6x4. And it is this 6X4 
which makes up 24, the rtal difference of tlie errors. — The 
difference between a negative and positive result being the 
sum of the differences between each of them and no result. 
Tbus, if I gain 10s., 1 am richer to the amount of 24*. than if 
1 lose li.*. 



^ 



rosiTiuN. 



353 



ilo, and M 



t o))tainfMl. 
fc required. 

of execs'^. 

im is 14+ 
era is 25 — 



ors. Then 



do number 



on 13 



to bo 



the results 
iiubers (tlio 
3 residt ob- 
lat obtained 
e difference 
ace between 
\ii numbers 
lumber, and 
aced in the 
at assumed 

mbpvs used 
s. For tlie 
)etween the 
, or both — 
. Tims, in 
;lt thau 19, 
been multi- 
is this 6x4 
rrors. — The 
. being the 
I no result. 
24«. than if 



t 



EXEUCISES. 

13. What number is it whit;li, boiur; niultipliod by 3, 
Uio product \n\\\\% increasod by 4, and the sum divided 
by 8, the quotieut will bo \V2 ? Ans. 84. 

14. A4!ron asked liis fatlier how old lie was, and re- 
ceived the foUowin,-]; answer. Your ago is now J- of 

What are their 



mine, but 5 years ago it was only i. 
ag(\s } Ans. 8;) and 20 
' 15. A workman was hired for 30 days at 2s. Qd. for 
every day ho worked, but with this condition, thafc^ for 
every day he did not work, he should forfeit a shilling. 
At the end of the- time he received £,2 14^., how many 
days did he work ? Am. 24. 

16. llcquired what number it is from which, if 34 
be taken, 3 times the remainder will exceed it by \ o^ 
itself .= Ans. 58-=. . 

17. A and 13 go out of a town by the same road. A 
g(jcis 8 miles each day ; 15 goes 1 mile the fir.st day, 
2 the second, 3 ' ' ' ' 
take A } 



the third, &c. When will B over- 



Suppose 



A. 
5 

8 

40 
15 



B. 
1 
2 
3 



5 



Suppose 



A. 

7 
8 

50 
28 



5)25 Is 



--a 

7 



35 

20 

1)15 



7)28 
-4 



5 



20 



B. 
1 
2 

• 1 
O 

4 
5 

G 
7 

28 



5 



4=1 



"Wc divide tho ciilive eri'nr by the number of daya iu each 
c.-.sc, -which gives tlie error iu one day. 

18. A gentleman hires two labourers; to the one bo 
gives M. each day; to the other, on the first day, 2(/., 
on the second day, Ad.^ on the third d;iy, 6^/., &.c. In 
how many days will they earn :tn e'pinl sum } Aim. In 8. 

11). What" are tho.s:j numhers whieh, when added, 



ij 



S54 



I'OSrTION. 



make 25 ; but when ono in halved and the other douhled, 
give (H[na[ results ? Ans. 20 aud 5. 

20. Two coutractoi-H, A and ]J, arc each to hiiild a 
wall of equal dimcusioiis ; A (iniploys as many men as 
finish 22J perches hi a r'ay ; Ij employs the first day as 
many as finish G perches, the second as many as finish 
9, tlie third as many as finish 12, &c. In what time 
will they have built an e(|ual number of perches ? 
Ans. In 12 days. 

21. What is vhat number whose ^, i, and -}, multi- 
plied together, make 24 ? 



Sujtposo 12 



1=3 

rroduct=i8 

3 41 

81 result obtained. 
2-1 result rc(iuircd. 

+57 

04, the cube of 4. 

3648, product. 

57+21=78 



Suppose 4 



Product=r;2 
a — U 



3 result obtained. 
24 result required. 

1728, the cube of 12. 



30288 To this pruduct 
3018 is added. 



5< 



7-21=78. 



78 )391)3 is the sum. 
And 512 tlio quotient. 



-3/512=8, is the required number. 

We nmltiply the alternate error by the cube of tlie supposed 
tmmbcr, because tlui errors belong to the g'^th part of the cube 
of tlie assumed numbers, and not to tlic nuinbei'S tlicniselvos ; 
for, in reality, it is the cube of some number that is required 
—tt'ince, 8 being hssumed, according to the question we have 

22. What number is it whose 1, J-, ]-, and 1, multi- 
plied together, will produce 699S| .'^ Am. 36.^ 

23. A said to B, give me one of your shillings, and 
I shall have twice as many as you will have left. B 
answered, if you give me Is., I shall have as many as 
you. Ilow ninny bird each } Avs. A 7, and B 5. 



POSITION. 



355 



24. There are two nuinbors wliich, when ailuo(1 to* 
gather, iiialca 30; but the J, J, and j, of the greater 
arv cqijal to |, a^ and |^, of the lesser. What are they ? 
Ans. Vj and Ijs. 

2' A f^ontlomaii has 2 horsoa and a sacMlo worth 
j£50. Tbo ,s:iddlo, it' set ' n th* baek of the first linrse, 
will make his value doubic that of the second ; but if 
set on the baek of the second horse, it will make his 
value treble that of the first. AVhat is the value of 
each horse ? j v. £30 and iJlO. 

2C>. A gentleman finding sov ral ben;£^ars at his door, 
pavt! to eii;h 4d. and had Gd. left, but if he had given 
ikl. to each, he would have had 12d. too little. How 
many bop:gars were thure ? Aiis. 9. 

It is so likely tliat those ) are desirous of stud^inr; 
this subject further will be acquainted with the method 
of troatin<5 algebraic equations — which in miiiy case? 
nffords a so much simpler and easier mode of solvin,'; 
qu.;stions belonging to position — that we do not deem 
it necessary to enter further into it. 

QUESTrONS. 

1. What is the diiTerence between single and double 
position.? [32]. 

2. In what cases may we expect an exact answer by 
ihesc rules r [32 j . 

3. Whtit is the rule for sin<>-le position ? [33] . 

4. AVhat are the rules for double position .'* [35 and 
40 j. 

MISCELLANEOUS EXERCLSES. 

1. A father being disked by his son how old he was; 
voplied, your age is now ^ of mine ; but 4 years ago 
\i was only ^ of what mine is now ; what is the ago of 
each } A as. 70 and 14. 

2. Find two numlxn-s, the dilForence of which is 30, 
nnd the relation between them as 7^ is to 3^.'' Am. 
58 and 28. 

3. Find two numbers whose sum and product are 
equal, neither of (hem being 2 ' Ans. 10 and 1^. 



.% 









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1.0 



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IM 

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Corporation 



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(716) 872-4503 




,,-y^': 



o 




356 



EXERCISES. 



What is the t;um of tlie series -J-, i, }, &c. ? Ans. 1. 



4. A^porson being asked the hour of tlio day, answered, 
It is between 5 and G, and 1;otli the hour and minuto 
hands are together. lle(iuired what it was ? Ana. 
27 f\ minutes past 5. 

5. 

6. What is the sum of the .series' |,'y\, j%y j-'A) ^^- • 
Ans. If 

7. A person had a salary of £75 a year, and let it 
remain unpaid for 17 years. How much had he to 
receive at the end of that time, allowing 6 per cent, 
per annum compound mtercst, payable half-yearly ? 
Ans. £204 17s. lO^d. 

8. Divide 20 into two such pav^s as that, when tho" 
greater is divided by the less, and the loss by tlie greater, 
and the greater quotient is multiplied ))y 4, and the less 
by 64, the products shall be equal.? Ans. 4 and IG. 

9. Divide 21 into two such parts, as that when the 
less is divided by the greater, and the greater by the 
loi'S, and the greater quotient is multiplied by 5, and 
ih" less by 125, the products shall be e(pial ? Ans. 
3'- and 171. 

1" A, B, and C, can finish a piece of work in 10 
days; }> and C will do it in 16 days. In what time will 
A do it by himself? Ans. 26| days. 

1. A can trench a garden in 10 days, B in 12, and 
in 14. In what time will it be done by the three if 
thoy work together ? Ams. In 3-,-Yt ^'-^J^- 

12. What number is it which, divided by 16, will 
leave 3 ; but which, divided by 9, will leave 4 ? Aiis. 
67 

i3. What number is it which, divided by 7, will 
leave 4; but divided by 4, will leave 2 ? Ans. IS. 

14. If £100, put to interest at a certain rate, wih, 
at the end of 3 years, be augmented to £115'7G25 
(compound hitercst being allowed), what principal and 
interest will hi due at the end of the first year ? Ans. 
£105. 

15. An elderly person in trade, desirous of a little 
respite, pi'oposcs to admit a sober, and industrious young 
person to a share in the business ; and to encourage 
him, ho olfors, that if hi^ circumstances allow him to 



F,XERCrSE!5. 



357 



advance £100, his salary shall be £40 a year ; that if 
ho is able to advance £200, he shall have £55 ; but 
that it he can advance £300, he shall receive £70 
annually. In this proposal, what was allowed for his 
attendance simply ? Am. £25 a year. 

16. If 6 apples and 7 pears cost 33 pence, and 10 
apples and 8 poars 44 pence, what is the price of one 
apple and one pear .?* Ans. 2d. is the price of an apple, 
and 3d. of a pear. 

17. Find three such numbers as that the first and I 
the sum of the other two, the second and i the sum of 
the other two, the third and \ the sum of the other 
two will make 34 ? A^is. 10, 22, 26. 

18. Find a number, to which, if you add 1, the sum 
will be d"visible by 3 ; but if you add 3, the sum will 
be divLsiVie by 4 ^ Am. 17. 

19. A market woman bought a certain number of 
eggs, at two a penny, and r,s many more at 3 a penny ; 
and having sold them all at the rate of five for 2^., she 
found she had lost fourpence. How many eggs did she 
buy .? Am. 240. 

20. A person was desirous of giving 3d. a piece to 
some beggars, but found he had 8^. too little ; he there- 
fore gave each of them 2d., and had then Sd. remain- 

llequired the number of beggars.? Am. 11. 

21. A servant agreed to live with his master for £8 
a year, and a suit of clothes. But being turned out 
at the end of 7 months, he received only £2 135. 4d. 
and the suit of clothes ; what was its value } Am, 

_ 16.9. 

22. There is a number, consisting of two places of 
figures, which is equal to four times the sum of its 
dtgits, and if 18 be added to it, its digits will be in- 
verted. Wiiat is the number ? Am. 24. 

23. Divide the number 10 into three such parts, that 
if the first is multiplied by 2, the second by 3, and the 
third by 4, the three products will be equal .? Am. 

24. Divide the number 90 into four such parts that, 
If the first i;^ increased by 2, the second dhninished by 
2, the third multipli(!d by 2, and the fourth divided by 



mg 



358 



EXERCISES. 



% 



2, the sum, clIfForcncc, product, and quotient will bo 
equal : Ans. 18, 22, 10, 40. 

25. \Vliat fraction is that, to the numerator of which, 
if ] is added, its viluo will be i ; but if 1 bo added to 
tHe denominator, its value will be •} ? Ans. j%. 

2^3. 21 gallons were drawn out of a cask of wine, 
which had leaked away a third part, and the cask 
being then guaged, was found to be half full. How 
much did it hold ? Ans. 126 gallons. 

27. There is a number, ^ of which, being divided by 
6, I of it by 4, and J- of 'it by 3, each quotient will 
be 9 ? Ans. 108. 

28. Having counted my books, I found that when I 
multiplied together i, j, and f of their number, the 
product was 162000. How many had I ? Ans. 120. 

29. Find the sum of the series l+'^-f j + |, &o. .? 
Ans. 2. 

30. A can build a wall in 12 days, by getting 2 days' 
assistance from B ; and B can build it in 8 days, by 
getting 4 days' assistance from A. In what time will 
both together build it ? Ans. In 6f days. 

31. A and B can perform a pisce of work in 8 days, 
when the days are 12 hours long ; A, by himself, can 
do it in 12 days, of 16 hours each. In how many days 
of 14 liours long will B do it } Ans. 13^. 

32. in a mixture of spirits and water, | of the whole 
plus 25 gallons was spirits, but i of the whole minus 5. 
gallons was water. How many gallons were there of 
each } Ans. 85 of spirits, and 35 of water. 

33. A person passed } of his age in childhood, yV of 
it in youth, | of it +5 years in matrimony ; he had 
then a son whom he survived 4 years, and who reached 
only i the age of his father. At what age did this per- 
son die ? Ans. At the age of 84. 

34. What number is that whose i ejjcecds its \ by 
72 ? Ans. 540. ^ _ . 

35. A vintner has a vessel of wine containing 500 
gallons ; drawing 50 gallons, he tlicn fills up the cask 
with water. After doing this five times, how much 
wine and how much water are in the cask.? Ans 
295^j)_ gallons of wine, and 204 J |i gallons of water. 



'' 



! , 



ni 



EXERCISES. 



350 



45. A mother an<l two daugliters working together 
nil 3 lb of fliix in one day ; the mother, by herself, 



ian do it in 2i days ; and the eklest daughter m ^j 
days. In what time can the youngest do it.? Ans. 

In f')— davs. 

37^ A merchant loads two vessels, A and B ; into 
A he puts 150 hogsheads of wine, and into B 240 hogs- 
heads. The ships, having to pny toll, A gives 1 hogs- 

15 <.;ivos 1 hogshead and 3().s\ 
iich hogshead valued ? 



head, and receives V2s. 

besides. At how much was c 

Ans. £4. 12.S-. , ^, . 

38. Tlireo merchants traffic in company, and their 
stock is i2400 ; the money of A continued in trade 5 
months, that of B six months, and that of G nine 
months; and they gained £^7b, which they divided 
cfpially. What stock did each put in.? Ans. AiilbT^-j, 

39. A fonntain has 4 cocks, A, B, C, and D,_and 
unilcr it stands a cistern, which can be filled by A lu G, 
by B in 8, by in 10, and by D in * .- hours ; the 
cistern has 4 cocks, E, F, 0, and II; and can be 
emptied by E in G, by F in 5, by Q in 4, and by II m 
3 hours. "Suppose the ci^^tern is full oi water, and that 
the S cocks are all open, in what time will it be emptied ? 
Ans. In2,^g hours. 

40. What is the value of -2^07' ? Ans. if 
41 What is the value of -541 G' ? Ans. Yi- 

42. What is the value of •0^7G923' t Ans. yV- 

43. There are" three fishermen. A, B, and C, who 
have each caught a certain number of fish ; when A's 
fish and B'sare put to:rother, they make 110 ; when 
B's and CVs are put together, they make 130; and when 
A's and C's are put together, they make 120. It the 
fish is divided equally among them, what will be each 
mairs share; and how many fish did each of them 
catch ? Ans. l<lach man had GO lor his share ; A caught 
50, B GO, and G 70. 

44. There is a golden cup valued at 70 crowns, and 
two heaps of crowns. The cup and first heap, are wortli 
4 times the value of the second heap ; but the cup and 



second heap, ai-e 



worth double the value of the first 



3r,o 



EXERCISES. 



heap. ITow many crowns arc there in eacli lieap ? Ana 
oO ill one, and 30 in another. 

45. A certain number of horse and foot soldiers ai'O 
to be ferried over a river ; and tliey agree to pay 2^d. 
for two horse, and Sid. for seven foot soldiers ; seven 
foot always followed two horse soldiers ; and when they 
were 'dl over, the ferryman received ^£25. How many 
horse and foot soldiers were there ? Ans. 2000 horse, 
and 7000 foot. 

46. The hour and minute hands of a watch arc' to- 
gether at 12 ; when will they be together again ? Ans. 
at 5/,- minutes past 1 o'clock. 

47. A and 13 are at opposite sides of a wood 135 
fathoms in compass. They begin to go round it, in the 
Same direction, and a* the same time ; A goes at the 
rate of 1 1 ftithoms in 2 minutes, and B at that of 17 
in 3 minutes. How many rounds will each make, before 
one overtakes the other ? Ans. A wiU go 17, and 13 
16J-. 

48. A, B, and 0, start at the same time, from the 
(same point, and in tlie same direction, round an island 
73 miles in circumference ; A goes at the rate of 6, 
B at the rate of 10, and C at the rate of 16 miles per 
day ^ In what time will they be all together again .^ 
Ans. in 36|- days 



IX 






M-ATIIEMATICAL TABLES 



LOGATUTIIMS OF NUMBERS FllOM 1 TO 10,000, WITH 
DIFFKIIENCES AND PROPOllTIONAL I'AllTS. 



I ' 





















1 




m^^M 


Numbers f 


•om 1 to 100. 


# 


I 


No. Log. j No. 


Log. 


NO. 


Log. 


No. 


Log. 


No. 


Log. 


I 


1 


. 

O-O'JOOOO 


•21 


1-32'2'219 


41 


1-614784 


61 


1-785330 


81 


1-908486 


H 


■i 


0- 301030 


•22 


1-342443 


44 


1-043249 


64 


1-792394 


84 


1-913314 


1 


3 


0-477141 


-23 


1-361743 


13 


I -0334 .3 


63 


1-709341 


83 


1-919073 


4 


0-60-JOGO 


21 


1-380211 


U 


1-C43453 


64 


1-800180 


34 


1-9-24479 




1 
(i 


0- 690970 


45 

'26 


1-397040 


45 
46 


1-053413 


05 


1-814913 


35 


1-9-29419 




!jHH 


0'778151 


1-414973 


1-664758 


66 


1-819544 66 


1-9^4498 


H 




0-Sl.JOOJ 


•27 


1-431364 


47 


1-674093 


67 


1-346075 87 


1 9.39519 




^H 


8 


0-903000 


•23 


1-417158 


43 


1- 631-241 


6S 


1-834509 


83 


1-944483 


mH 


9 


0-9J4243 


29 


1 •462393 


49 


1-6901% 


69 


1-838349 


89 


1-949390 


■ 


10 


1-000000 


30 
31 


1-4771-21 


60 
51 


1-693970 


70 


1-845098 


90 


1-954413 


■ 


11 


l-0413ri3 


1-491364 


1-707570 


71 


1-851'253 


91 


1-959041 




M 


12 


1 -079181 


32 


1-505150 


53 


1-716003 


74 


1-8573.34 


92 


1-963783 


^H 


13 


1-113943 


33 


1-518514 


63 


1-7-24476 


73 


1-863343 


93 


1 -968433 




^H 


14 


l-UOlia 


31 


1-531479 


54 


1-732394 


74 


1-869434 


94 


1-973128 




iHI 


15 


1- 170091 


35 
36 


1-644063 


65 
56 


!■ 740303 

1 


75 


1-875061 


95 


1-977724 


^H 


IG 


l--2041-iO 


1-550303 


1-748188 


76 


1 -830314 


96 


1-98'2471 


m 


17 


1 --230449 


37 


1-533404 


57 


1 -766875 


77 


1-886491 


97 


1-986772 




H 


13 


\-2iio-2l'i 


33 


1 •5r9784 


53 


1-763143 


73 


1-894095 


93 


1-991226 




■ 


19 


1-478754 


39 


1-591005 


59 


1-770854 


79 


1 -SO? 647 


99 


1-995635 




■ 


•2ii 


I • 301030 40 


1-604060 


CO 


1-773151 


SO 


1 -903090 


100 


2-000000 


IB^^I 



"mm 



302 



rOGARITIIMP. 



r 1' 


100 


1 1 


3 1 3 1 4 


6 


6 


7 1 8 





1). 

132 




:oouuoooooi3i 


0008O.sOOI3ni|0017.34 UO'2IOO 


002.598'00.3029 00.3461 


003891 


II 


11 in-Jll 4751 


6181 660!)! fif38 6466 


68941 7321 1 7748 


8171 


128 


Hii 


•2 K000[ !)0-20 
3'01'JdH7 013'20!) 


9l,-.l !)n76 010300 010724 


011147i01l570011993;0l211.. 


121 


101 


tU30KO 014100 4.V21 


4910 


6360 


5779! 6197 


6616 


120 


Ifiti 


I 


7033 74')] 


7868 8284 8700 


9116 


9532 


9947 020361 


020775 


116 


■-'o; 


6 


0-2llH!)uai(i03 


022ai6]02'2428 022811 


023252 


023604 024075 4486 


4896 


112 


VilH 


G 


TMHy 571;-, 


61 •2.-. 


6533! f:942 


7350 


77.-)7 8164 8571 


'3978 


108 


•J!M) 


7 


'3' ll 978l'i 


03019.) 


030(ilU):031004|031 108 


031812 03-2216 032619 


0M02I 


101 


;-t;il 


/24',mi26 


42;;7 


4628, 6029 


5430 


6830 62301 6629 


702.M 


!0l» 


;);:) 


110 


',4:20 78-26 


8-223 


H620| 9017 


9414 


9811 040207,040602 


040998 


;J97 
.193 




(i;i3:)3 011787 


042l82|012.-)76 042909 


043302 


043755 044148 044.540 


0449.32 


;j^ 


1 


;':.3-23j 6714 


OlOril 6I9.-)I 688.'. 


7275 


7064 


8053 8442 


8830 


390 


7li 


• 1 


9218 iUiOli 


9993 J0380'n.'.070fi 


051153 


051538 


051924 0.52309 


0.5-2(i91 


386 


li:i 


3 


003('7S (l;-(3403 


053846 4230 461.1 


4;)96 


5378 


,OT60 6112 


6521 


.■i83 


1.^1 


4 


UDO/)! 7-2b0 


7006 8046 8126 


8805 


9IH5 


9563 9942 


060320 


.179 


iy:< 


t 


OGO()!I800107') 


Oai4.>2 061829 062200 0(!2J82 


002958 


063333 063709 


4083 


376 


•M^ 


6 


4i;)S 483-2 


520(1 ooSn 6!»53| 6320 


6699 


7071 1 7443 


7815 


373 


•]«.-, 


7 


81Si)i 8;)i")7 


8928 9298 UtiUS 07003S 


070407 


070776,071115 


071611 


370 


aoo 


H 


0718+2 07U^2;")0 


072617 07298.-, 0733..2 3718 


4085 


4151 


4816 


5182 


366 


:M() 


lyo 


.');:.47 .01)12 


6276 0640 7004 7308 


7731 


8094 


8457 


8819 


163 
:J60 




079181 07l)")43 


0799U4;080200'080ll26 


080937 


081347 


081707 


082067 


08212(; 


3;-. 


1 


08278.-., 083141 


083.-.03 3861 


4219 


4576 


493 1 


5291 


5647 


6001 


357 


70 


o 


0;l!iiH 0710 


707 1 712{i 


7781 


8136 


8490 


88 15 


9198 


9552 


355 


101 


3 


990;V090-208 


09061 1]090903 


091315 09 I6(i7 


09-2018 


092370 


092721 


093071 


:J52 


i;i<i 


4 


093t-2-2J 3772 


4122 4471 


4-20 


5169 


5513 


rySGii 


6215 


656^ 


.549 


17-1 


6 


091(1 ~-2:u 


7604 79.-,l 


8298 


864 I 


8990 


9335 


9681 


10002(i 


346 


20;i 





100371' 10071.". 


1010;;9 10! 103 


101717 


10'2091 


10-2434 


102777 


103119 


3162 


.113 


■211 


7 


3804 4140 


4 187 


482 S 


6169 


6510 


5851 


6191 


6531 


6871 


:t41 


•J7m 


H 


7210 7649 


78.'-8 


8-227 


8565 


8903 


9241 


9,)79 


9916 


110253 


138 


■M.i 


9 


110o90,1109'2G 


111203 lll.)99|111934J112-270 


112605 


U2940 


113275 


3609 


335 
433 




130 


I!3s;i3;n4^277 114611 U4')44;ll;V278'lli)^ir, 


115943 


116-276 116608 


11 694 f) 


;;o 


1 


727 ll 7003 793; 


8265 8595 8926 


9256 


9586 


9915 


120-24.-i 


330 


01 


•2 


1-20.-.74' 1-20903 1-2I-23I 


1215(i0 121888 122216 


122544 


122871 


123103 


3525 


■',-lS 


!»; 


A 


3^iV2 4178 4.)0t 


4830 


5150 6481 


5806 


6131 


6451' 


6781 


325 


i-ii) 


4 


710.-. 71^vft 77 -.3 


8070 


8399 8722 


9045 


9368 


9690 


130012 


3-23 


101 


.'. 


1.30334 130tJ.>;) 130977113^298 


131619 131939 


132260 


132580! 132900 


3219 


.!21 


IPS 





3r.39 3858 4I77| 41;)o 
0721 7037 7354I 7071 


48141 01.33 


5451 


6709 6086 


6403 


.(18 


■s-i-') 


1 


7987 1 8303 


8618 


8934 9219 


9561 


.il6 


\!..b 


8 


9879 140191 


140.-;03 


140822 


141136 141450 141703 142076'142389 


14270-2 


:il 


i!)U 


9 


11301.) 


3327 


3639 


3951 


42631 4571 4835 6196 5507 


5811- 


ill 

:!09 




llu 


140r23 


140 138 


14074S| 147058 


147.3671147670 147985 148-294 148603 


148911 


30 


1 


9^2 1 9 9rv27 


9835|150112|I50119;1507j6|151063;151.370;1o1676;15198-. 


:(07 


(M 


o 


\ry2-i^» l;V2r>94 


152900 3205 


3510 


3815 


4120 


4421 


4728 


.503'2 


.(05 


!)l) 


«J 


;)330 


5040 


5943 C2I0 


0549 


6855 


7154 


74.57 


7759 


8061 


.'.03 


IJO 


4 


830^2 


8004 


890--) 92()0 


9567 


9868 


160108 


160469 


160769 


16106b 


:!ni 


1 oO 


-• 


101308 


161607 


161967 


162200 


162564:162863 


3161 


3460 


37.58 


4055 


299 


180 





43"<3 


4Q,jO 


4947 


5214 


5541 6838 


61.34 


6430 


(i726 


702^- 


297 


■J 10 


7 


7317 


7013 


7908 


8203 


8-197 8792 


9086 


93 SO 


9674 


990; 


295 


210 


fi 


170202 170.-i,-)0;i70818 171141 


171434 171726 172019:17231 11172603 


172895 


J93 


■270 


9 

i;.o 


3180| 3478 3709 4000 


4351 


4641 4932 6222 6512 


680-j 


.'91 

2.S9 




170091 170381 Il700ro' 170969 177^i4.9 


177536:177.825' 17811 3 178 lul 


1786,Si' 


•js 


1 


8977! 9-2041 9-;.V2l !i839;i801-2t):i80413:i80690'l809SG 181-27-2'l816.>i- 


287 


c')0 


■) 


I.-'I814!I82129 


182415:182700 2985 


32701 .3555 3!s39 4123 440-. 


.'85 


81 


3 


40.1)1 497 J 


5259 6542 6S25 


61(»8 0391 0674 6956 723f' 


.■83 


irj 


4 


7.-.21' 7803 


.808 1 83001 8647 


89^18 9209 9490 9771 190051 


281 


110 


;') 


iyU33-i;lH(l()l:» 


190892.191171 


191451 !l917.30192010 192289:192507 28 li: 


;79 


llirf 





312..| 340.; 


308 1 3>)5;i 


•1237 4,-.ll 4792! 5.)(;9 5313 5i;-2: 


.'78 


l'.)u 


7 


.'.900 6170 


6453! ('72:> 


7005 7-!8l 7.V.6' 7832 .SI 07 .'-;i8. 


;T(i 


^■24 


K 


8(i..7 >-93'2 


92(16: 9!8l 


9755 2000-29 200303 209577 20(W50;2Ol 12 1 


.'74 


:2^<-2 


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20(39; ;201tihi,J0i:M.-i-20^2J10/20-j488| :;2701| 30331 330i)| 3u77| 3.'.l.- 


.^72 



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|-;, 

7 

10 

1 '■' 
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lOM 


21 


101 


>.s 


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.i!i:{ 


10 


:!0() 


11 


:?:iii 


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:)i^;! 





.17!) 


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:i7.i 


T) 


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3 214188 
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1632 


1680 


1629 


1677 


1726 


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49 


34 


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1823 


1872 


1920 


1969 


2017 


2066 


2114 


2163 


2211 


2260 


48 


29 


6 


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2366 


2105 


2463 


2602 


2660 


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2617 


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48 


34 


7 


2792 


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2839 


2938 


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3034 


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3131 


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48 


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3326 


3373 


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3760 


3803 


3856 


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900 


961243i9,'>4291 


954339 


954387 


9.54436 


964434 


964632 


964680 


9546281964677 


5 


1 


4;2.-,i 4773 


4821 


4869 


4918 


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6014 


6062 


5110 


5158 


48 


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6496 


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5640 


48 


14 


o 


C-OS.S 


6736 


5784 


6832 


6880 


6,928 


6976 


6024 


6072 


6120 


48 


19 


4 


616S 


6216 


62G") 


6313 


63iil 


6 109 


6467 


6606 


6563 


6601 


48 


24 


6 


0iJ49 


6697 


6746 


6793 


6840 


6888 


6936 


6981 


7032 


7080 


48 


29 


6 


7128 


7176 


722 1 


7272 


7320 


7368 


7416 7464 


7612 


7659 


48 


34 


^ 
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7656 


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7761 


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7894 7942 


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43 


38 


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8086 


8134 


8181 


8229 


827 7 


8326 


83? 3 8421 


8468 


8616 


48 


43 


9 
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b)6l 
969U41 


8612 


S6.^9j 87071 8706 


8803 


88,i0 8898 


8946 


8994 


48 

48 


969089 


9,>91.37 969186 


969232 


959280 959328'969376 


969423 


9.59471 


5 


1 


9,. 18 


9>fl6 


9614 


9661 


9709 


97.67 


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9900 


99.17 


48 


9 


2 


i'ni'.)!} 960042 


969090 


9001.33 


960186 


960233 


960281 


960328 


960376 


960423 


48 


14 


3 


960171 0,-il8 


0666 


0613 


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0709 


0766 


0804 


0861 


0899 


48 


19 


4 


0916 


099 I 


10-11 


1089 


1136 


1184 


1231 


1279 


1326 


1.374 


47 


24 


6 


1121 


1469 


1616 


1663 


iOll 


16.-.8 


1706 


1763 


1801 


1348 


47 


28 


6 


1896 


1943 


1990 


2038 


2'i86 


2132 


2180 


2227 


2276 


2.322 


4'r 


33 


7 


2369 


2417 


2464 


2611 


2669 


2606 2663 


2701 


2748 


2795 


47 


38 


8 


2813 


2890 


2937 


293;) 


3032 


3079 


3126 


3174 


3221 


3268 


47 


42 


9 

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3310 


33G3 


3110 


34,67 


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3652 


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963788 


963836 


903882 


963929 


963977 964021 


961071 961118 


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4304 


4401 


4148 


4496 


4612 4690 


4637 


4681 


47 


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4731 4778 


4826 


4872 


4919 


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6013 6001 


6103 


6 1 56 


47 


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3 


6202 


6249 


6296 


6313 


6390 6137 


64,84 6631 


6.678 


6626 


47 


19 


4 


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6719 


6766 


6813 


5860 


5907 


6964 6001 


6048 


6096 


47 


23 


6 


6142 


0189 


6236 


6283 


6329 


6376 


6423 6170 


6617 


66('4 


47 


29 


6 


O.ill 


t;6,i8 


6706 


0762 


6799 


6346 


6892 6939 


6986 


7033 


47 


. 33 


7 


7080 


7127 


V173 


72-,;0 


7267 


7314 


7361 7408 


71,5! 


7.501 


47 


38 


8 


7,348 7,i96 


7642 


7 688 


7736 


7782, 78291 7876 


7922 


7969 


47 


42 


9 


80 1 6 S,)62 


8109 


8166 


8203 824:-){ 8296; 8313 


8390 


8436 


47 


9684831968630 


96,'^,;76 968623 


968670|96-i7i6'968763 96881 


968866 


968903 


47 


6 


1 


MV,0 8996 


9043 


9090 


OUlol 9183 


9229 9276 


9323 


9369 


47 


9 


2 


9116 9163 


9609 


9660 


9602 9649 


9696 9742 


9789 


9836 


47 


14 


3 


9382 9928 


9976 


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970068 97011 ! 


970161970207 


970264 


970300 


47 


io 


4 


970347 


970393 


970440 


0486 


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06'j6; 0672 


0719 0766 


46 


29 


5 


0812 


0368 


0904 


0951 


0997 


1044 


1090 1137 


1183 


1229 


40 


23 


6 


1276 


1.322 


1369 


1416 


1161 


1.608 


1. 6.541 1601 


1647 


1693 


46 


32 


7 


1740 


1786 


1332 


1879 


1926 


1971 


2018 2064 


2110 


2167 


46 


37 


8 


2203 


2249 


2296 


2342 


2388 


2134 


2481 2527 


2.573 


2619 


46 


41 


9 


2666 


271-2 


2768 


2804 


2861 


2397 


2943 2989 


3035 


30t.2 


46 



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iii^ 



. . 



i 1 




,t I 



J76 



LOOARITHMS. 



5 



14 

IH 

37 
41 



6 
9 
14 
18 
2:» 
•i7 
3-2 
3t< 
41 



N. 

340 
1 
2 
3 
4 



900 
J 

3 

4 
6 
6 

7 



9731-iS 
30D0 
4051 
451i 
4972 
f)432 
5891 
63f)0 

fisoa 
72aa 



1)77724 
8181 
8037 
9093 
9548 

980003 
0458 
0912 
13fifi 
1819 



1 



973174 
3()36 
4097 j 
45581 
60181 
6478 
6937 
6396 
6854 
7312 



973220 
3f.82 
4143 
4604 
6064 
6524 
6983 
6442 
6900 
7358 



973266 
3728 
4189 
4660 
6110 
5570 
6029 
6488 
6946 
7403 



977769 
8926 
86u3 
9138 
9594 

980049 
0503 
0957 
1411 
1864 



977815 
8272 
8728 
9184 
9639 

980094 
0549 
1003 
1456 
1909 



977861 
8317 
8774 
9230 
9685 

980140 
0694 
1048 
1501 
1954 



973313 
3771 
4235 
4696 
5156 
5616 
6076 
6533 
6992 
7449 




977906 
8363 
8819 
9275 
9730 

980185 

0610 

1093 

164' 

2000 



977952 
0409 
8865 
9321 
9776 

980231 
0685 
1139 
1592 
2045 



977998 
8454 
8911 
9366 
9821 

980276 
0730 
1184 
1637 
2090 



982452 



2904 
3356 

3307 
4257 
4707 
6157 
6606 
6055 
6503 



982497 
2949 
3401 
3852 
4302 
4752 
6202 
6651 
6100 
6548 



8 



9 



973369,973405 97345 1 973497 

3820 3866 3913 3959 

4281 4327 4374 4420 

4742 47 88 4834 4880 

5202 6218 6294 6340 

6062 6707 6763 6799 

6121 6167 6212 6258 

6579 6626 6671 6717 

7037 7083 7129 7175 

7495 7541 7686 7632 



978043 
8500 
8956 
9412 
9867 

980322 
0776 
1229 
1683 
2136 



973543 

4005 
4166 
4926 
6386 
5845 
6304 
6763 
7'?20 
7678 



978089 
8546 
9002 
9457 
9912 

980307 
0821 
1275 
1728 
2181 



9SG951 
7398 
7845 
8291 
8737 
91 S3 
9628 
|990072 
j 0516 
0960 



982543 
2994 
3446 
3897 
4347 
4797 
6247 
5696 
6144 
6593 



982583 
3040 
3491 
3942 
4392 
4812 
6292 
5741 
6189 
6637 



982633 
3085 
3536 
3987 
4137 
4887 
8337 
6786 
6234 
668i 



978136 
8591 
9047 
9503 
9968 

980412 
0867 
1320 
1773 
3226 

982678 
3130 
3581 
4032 
4482 
4932 
6382 
6830 
6279 
6727 



936996 987040; 



7443 
7890 j 
8330 
8782 
9227 
9072 
990117 
0561 
1004 



748S 
7934 
b;>jl 
(■^S20 
9272 
9717 
990UU 
0605 
1049 



9870S5 
753« 
7979 
8425 
837 1 
9316 
9761 

990206 
0650 
1093 



1802 


1846 


1890 


2244 


2283 


2333 


2686 


2730 


2774 


3127 


3172 


3216 


3568 


3613 


3657 


4009 


4053 


4097 


4449 


4493 


4537 


43B9 


4933 


4977 


5328 


6372 


5416 



937130 
7577 
8024 
8470 
8916 
9361 
9306 

990250 
0694 
1137 



45 
45 
46 
45 
45 
45 
45 
45 
45 
45 



937176 
7622 
8068 
8514 
8960 
9406 
9360 

990294 
0738 
118! 



5 991359 iiOl 103,991448991492 



1935 
2377 
2819 
3200 
3701 
4141 
4531 
5021 
5460 



67 995811 995854 995898 



6337 
6774 
7212 



6205 


6249 


6293 


6643 


6637 


6731 


7030 


7124 


7168 


7517 


7501 


7005 


7954 


7998 


8041 


8390 


8434 


8477 


8326 


B869 


8913 


9261 


9305 


9348 


9096 


9739 


9783 



8086 
8521 
8956 
9392 
0826 



991536 
1979 
2421 
2363 
3304 
3745 
4185 
4625 
5065 
5504 



995942 
6380 
6818 
7255 
7C:)2 
8129 
8564 
9000 
9435 
9870 



991530 
2023 
2165 
200 
3343 
3739 
4229 
4669 
6103 
654' 

1 

995986 
6424 
6662 
7299 
7"36 
8172 
8608 
9043 
9479 
99i8 



45 
45 
45 
45 
45 
45 
44 
44 
44 
44 



991626 
2067 
2609 
2901 
3392 
3833 
4273 
4713 
6152 
6591 



996030 
6468 
6906 
7343 
7779 
8216 
8652 
903- 
9522 
9967 



44 
44 
44 
44 
•11 
44 
44 
44 
44 
43 






A TABLE OF S(iUAHKS, tj0hK9, AND ROOTS. 



377 



1 


9 


D. 


n 


73.543 


46 




4005 


46 




41t)b 


46 




4926 


46 




6386 


46 




5846 


46 




6304 


46 




6763 


46 




7'?20 


46 




7678 


46 


i 


)78136 


46 




8591 


46 


> 


9047 


46 




9503 


46 


I 


9908 


46 


r < 


)80412 


46 




0867 


46 


> 


1320 


45 


i 


1773 


46 


1 


3226 


46 


J 


982678 


45 


5 


3130 


46 


i 


3581 


46 


? 


4032 


45 


7 


4482 


46 


7 


4932 


46 


7 


6382 


45 


6 


6930 


46 


4 


6.i79 


45 


2 


6727 


46 





987176 


45 


7 


7622 


45 


4 


8068 


45 


(» 


8514 


48 


6 


8960 


45 


1 


9406 


45 


6 


9360 


44 





990294 


44 


4 


0738 


44 


7 


1182 


44 


to 


991626 


44 


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2067 


44 


>5 


260S 


44 


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2901 


44 


H 


3393 


44 


3.0 


3833 


44 


29 


427f 


44 


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44 


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61oV 


44 


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5591 


44 


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99603( 


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No. 


flqonre. 


Cut. p. 


Sq. Root. Cuho Iloot 


No. 
64 


Sqiiaru 


C'tilie. 


.^q. Root. 


Cube Root 




I 


1 


1 


1-0000000 1 000000 


4096 


262144 


8-0000001) 


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a 


4 


8 


1-4142136 1-260021 


66 


4-226 


274626 


8-06-2-2677 


1-020726 




3 


9 


27 


l-7320.')08 1-412260 


66 


4366 


287496 


8-1240331 


4-041240 




4 


16 


64 


2-0000000 1-687401 


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4489 


300763 


8-1863628 


4-0616-18 




6 


25 


125 


2-2360680 1 -709976 


68 


4624 


314432 


8-2462113 


4-081656 




6 


36 


216 


2-4494897 1-817121 


69 


4«61 


328609 


8-2066-239 


4-101660 




7 


49 


313 


2-6467613 1-912931 


70 


4^)00 


343000 


8-3666003 


1-121286 




8 


64 


512 


2-8284271 i- 000000 


71 


6041 


367911 


8-4-261498 


4-140318 




9 


81 


729 


3-0000000 2-080084 


72 


6184 


373248 


8-48.5-2814 


1-160168 


.. 


10 


100 


1000 


3-102-2777 2-164136 


73 


6329 


389017 


8-6440037 


4-179339 




11 


121 


1331 


3-3106-248 2-223980 


74 


547<) 


40.5224 


8-60-23-263 


4-193336 




12 


144 


172S 


3-4011016 2-289128 


76 


5626 


421876 


8-6602640 


1-217163 




13 


169 


2197 


3 -6066513 2 -361. 3;<6 


76 


6776 


438976 


8-7177979 


4--i368J4 




14 


196 


2744 


3-7410574 2-410142 


77 


6929 


466633 


8-7749644 


4-254321 




16 


226 


3376 


3-87-29833 2 -466212 


73 


6084 


474562 


8-8317609 


■1-272669 




16 


266 


4096 


4-0000000 2-51984-2 


79 


6-241 


493039 


8-0881944 


4-290841 




17 


289 


4913 


4-1231066 2-571282 


80 


6100 


61-2000 


8-944-2719 


4-308870 




IS 


S54 


6832 


4-2420407'2-62'>741 


81 


6661 


631411 


9-0000000 


4 -.326749 




19 


861 


6859 


4-3688939 2-608402 


82 


67-24 


661368 


9-0663861 


4-344481 




20 


400 


8000 


4-4721300 2-714418 


83 


6889 


671787 


9-11043.36 


4-36-2071 




21 


441 


9201 


4-6826767,2-763924 


84 

8,^ 


7066 


692704 


9-1661614 


4-379619 




22 


484 


10048 


4- 09041 58'2- 80-3039 


7226 


014126 


9-219.6416 


4 -.396830 




23 


629 


12167 


4-796831612-843867 


86 


7396 


636066 


9-2736186 


4-414006 




24 


676 


13824 


4 -8989796 ,2 -334499 


8T 


7669 


668603 


9-3-27.3791 


4-431047 




26 


026 


15626 


&-0000000!2-9-24018 


83 


7744 


681472 


9-380.-3316 


4-447960 




26 


676 


17676 


6-0990196;2-9d2496 


y:) 


7921 


704969 


9-4339811 


4-464746 




27 


729 


19633 


5-19615-24 3-000000 


90 


8100 


7-29000 


9-436.3330 


4-481406 




20 


784 


21962 


6 -29160-26'3- 036689 


91 


8281 


7.63671 


9-6.393920 


4-497941 




29 


841 


24389 


6-3361648 3-072.'il7 


92 


8461 


778688 


9 -.69 16630 


4-514367 




30 


900 


27000 


6-4772266 3-107'232 


93 


8619 


804367 


9-6436608 


4-530656 




31 


961 


29791 


6-6677644 3- 141.381 


94 


S336 


830584 


9-6953697 


4-6468.36 




32 


102{ 


32768 


6-6668612 3-174802 


96 


9026 


867376 


9-7467943 


4-662903 


■ 


33 


1089 


36937 


5 -74 166-26 3- -207634 


96 


9216 


884730 


9-7979.590 


4-578867 


! 


34 


1166 


39304 


6-8309619 3-23961-J 


97 


9409 


912073 


9-8488678 


1-594701 




85 


1226 


4287;-. 


6-9160798 3-271006 


93 


9604 


94119-2 


9-8991949 


4-610436 




.S6 


1296 


46666 


6-0000000:3-3019-27 


99 


9801 


970-299 


9-949374 1 


4 - 626066 




.Hi 


1361) 


bmr,.\ 


6 -0327026;3- 3322-22 


100 


1 0000 


1000000 


10-0000000 4-641689 




3S 


1444 


64872 


0-1614140,3-30197;> 


101 


10201 


1030301 


10-0498766 4-667010 


:!0 


1621 


69319 


6- 244 9980:3-39 1211 


102 


10401 


1061208 


10-0996019!4-67-2329 




40 


KiOO 


64000 


6 -32 15663 3-419962 


103 


10609 


1092727 


10-1488910 4-687643 




■!1 


Hi 3 1 


68921 


6-4031242:3-418217 


104 


10:ilt) 


1124861 


10- 19.303904 -702669 




42 


1764 


7408d 


6-4807 107'3-47C02V 


106 


110-26 


1167li2.5 


10 -2469,)03!4- 717694 




43 


1849 


79607 


6-667438613-603398 


106 


11-236 


1191010 


10 -29.56301 ,4 -73-2624 




44 


1936 


861. HI 


6 -633249ij!3- 630348 


107 


11149 


1226043 


10-3440804:4-747469 




\> 


2026 


91125 


6-708203;)l3-666y93 


108 


11664 


1269712 


10-392304«,4-702203 




• UJ 


21 Id 


9; .-WO 


0-7823300i3-6830Ks 


109 


llsSl 


12950-29 


10- 44030o6'4- 7768.06 


■' 


4< 


22i.i'J 


10;5823 


6-8666546!3-60c;8-20 


110 


12100 


1331000 


I0-48808o6i4-7914-20 




•1.-! 


2304 


\W„'H 


6 --.'282032 '3 -634241 


111 


12321 


1367631 


10-6366.533:4-805396 




49 


2101 


117619 


7-0000000 3-669300 


112 


12641 


1404928 


10 -.60300.02:4 -820-284 




60 


2600 


12u000 


7 -07 10678:3- 6840.) 1 


113 


12769 


1442897 


i0-630!463:4-834.6S8 




61 


200 1 


132661 


7-U11284!3-7081:;o 


il4 


12996 


1481614 


10-6770783 4-848808 


- 


62 


2704 


I40o0w 


7-2111026:3-73-2611 


116 


13226 


1,520876 


10-7233063 4-862944 




b3 


2;:0a 


Mo.ii? 


7-2301099;3-76628i) 


116 


13466 


1660896 


10 - 77, ;3296 4-870999 


-! 


,j4 


2;-il6 


16,4o4 


7-3484690 3-779763 


117 


1063'.) 


1601613 


10-816:) 538 4-890973 




66 


3U2;i 


16637) 


7-41619Su!3-80-2963 


118 


13924 


hi 4303-2 


10-8627306 4-904368 


A 


.i^> 


3i:>6 


176616 


7-48331 is;3- 826802 


119 


i-lOl 


168^169 


10-908712!:4-918686 




.)7 


3249 


l.>361.93 


7-6498341;3-843.601 


1--.H) 


1 ! 100 


1728000 


10-9644612 4-9324-24 


k 


;i:H 


33(jt 


!9.M)-J 


7-fi!6773!!3-.870877 


i-2' 


1 4641 


177166! 


11 -0000000 4- 94:>0a3 


> 


69 


34t;l 


2063;9 


7 -681 14-57 Is -Ly 2900 


122 


1488 4 


1816848 


11-01.53610 4-969676 


1 


lit; 


3G00 


216000 


7-746966713-91 1867 


123 


16129 


1 860867 


11-0110636)4-973190 


1 


(5! 


3V2i 


226'J81 


7 -6102497 


3- 93649 r 
3-967802 


124 


16;i76 


I;i0u624 


11-1366287 4-986631 


C2 


3844 


23.':!328 


7-8740079 


1-26 


1602o 


1963126 


11-1303399,6-000000 


? 


63 


3989 


260017 


7-9372539 


3-9790.57 


126 


1637f 


2000376 


ll-2-24972'i,6 -01 3-298 

1 



i'i 



! 



l!f 



h 



J r; 



nil 

'I j I ' 
M i 



'I i 



! ( 



373 



SQUARES, CI UKS, AND U00T8. 



No. 



1Q7 



Sqiinrt.i C'lilic. 



130 

lai 

133 

131 

135 

i:<6 

137 

138 

139 

MO 

141 

142 

143 

144 

146 

146 

147 

148 

14!) 

160 

151 

162 

163 

154 

155 

ISO 

167 

158 

169 

160 

Itil 

162 

163 

104 
10.' 

166 

137 

168 

169 

170 

171 

172 

173 

174 

175 

176 

177 

178 

179 

ttiO 

181 

182 

183 

1 84 

18o! 

l8o| 

187. 

18H 

189 



1U120 

I63H4 

Itilitl 

16900 

17101 

17424 

176891 

179661 

18226 

184961 

187091 

19014 

193 Jl| 

196901 

198811 

20164! 

2(J44i» 

20736 

2102"; 

21310 

21009 

21004 

22201 

22.>00 

22?01 

23104 

2340!) 

23716 

240iJ.' 

24336 

24649 

24964 

36281 

20600 

26921 

26244 

26;j()9 

26896 

27226 

276)6! 

2788;) 

28224 

28061 

28900 

29241 

29684 

29929 

302Vii 

30026 

3ii!i.'6 

313J9 

3168i 

3J011 

3-'4iJ0 

32 ,"1)1 

331 



2048383 

2097162 

2146089 

2197000 

2248001 

2290968 

2362ti3 

2400 1 U4 

2lli03/., 

2616 i:o 

2671363 

2028072 

2686619 

2MU)00Ul 

2riO:U21'll 

286.1'28,si 1 1 



3:iis9 
33860 
34226 
34.".96 
3 1909 
36341 
36721 



I 



8q. lUiut. Ciitij lluul 



11 -2694277 16-02H620 
1 1-31 37086! 5- 0396S1 
ir367H167i.»0627;4 
ll-40i;643|6-066797 
11 -4456231 1 6- 0787 63 
ir4!i;91263|6-091613 



ll-o3266-.'6|6 
11-6768369 6 



104169 
117230 

■ 1-^9928 
1 12603 

•16613; 



29:^ 1207 
298,)98 4 
30 (8626 
3112136 
3176623 
32ll79i 
3307949 
337..000 
344296! 
3611808 
3681677 
366-2264 
3723876 
3796116 
386i)»93 
39443 1-i 
4019679 
40ll(i000 
4173281 
42616-28 
4330747 
4410944 
4492126 



6 189600 1 6 
601903h;6 
7046999:5 
7473414 6-167649 
7898261|6- 18010 
8321696;.)- 192494 
8713421 6 --204828 
9l637.")3!6--217103 

1 1 -968-260 7 1 6- 

12-0000000 6 

I 2-04169 lo! 6 

12-08.'J0460'6 



^39321 
211 183 

263688 
206 



N.» 



100 
191 
192 
193 
194 
196 
196 
197 
198 
199 
200 
201 
202 
-203 
■201 
20) 
06 
207 
208 



flquurc 



637|-209 
i2-1243.667|6'2776:i2l2' " 



16.V)-i6l|6-289.<72 
2066666 6 -.3014.-)9 



12-2474487 6-313-293 
1 2 -288206t;|6- 326074 
12-3-2882S0 16-336803 
12-3693169;6-318181 
l2-409673i)!6-36010d 
12 -4498996; 6 -37 1 686 
12-4899960 6-383213 
12-6299641 6-3i)4691 
1 2 -.6638061 6-406120 
12-6096-202, u-417601 
12-6401106 6-4-28836 
l-.'-6886776|6-44012 
12-7279221 i6-461362 
12-76714.63|6-462666 
12- 806-2486 ;6 



Cnbf. 



12-84.")232rti5 



46742..»6;12-8840987a 
4667433! 12- 9228430 15 
4M1032;12-961481 t 6 
4826809, 13 -0900000 '6 
4913000!i3-0384048"6 



47.370-1 

48481)0 
496866 
606879 
617818 
5287 7. '5 
639668 



6000211! 13 -076696»,6- 6^0 199 



■561-29,-; 
(7 206^1 



■ 604079 



5088448 13-1148770 
51777171 13-1629464 
526 J024| 1 3 - 1 90.9060 6 - ,)82770 
5369376 1 :', • -228766615 -f- 934 46 
6461776: 13--266 4992:6 
5.6462:j3J3-3041:)47i6 
66:i;l7.)2; i 3 ■ 34 1 664 1 6 - 62.)226 
6736339 1 3 - 3790882 5 - 6367 I i 
[>:;32O00 13-416 4(179 6-646216 
6929711 13-46:i62.10;6-6-)6661 
(;;)-28608' 1 3 - 4907376 6 - 667061 
6 l-2'U87j 13 -6-277493:6 -6771 11 
6229.JO 4' 1 3 - 6646600:6 - 687734 
63;'.1626l.'.-601 '17066 -OOoOlO 
6434866 13-638181715-708267 
6.j39203 13-0747943;6-7 18479 
66446 72: 13-711 3092;6 -728664 
0751269 13-7477-271|o-738794 



10 
!11 
212 
213 
214 
215 
216 
217 
213 
219 
220 
221 
222 
223 
224 
-225 
226 
2-27 
228 
■229 
230 
■231 i 
232l 
233 
234 
236 
236 
237 
2-W 
239 



-614673 210; 

24 1 ; 

242! 
243 j 
2441 
246 1 
246 
247! 
•248 1 
-249| 
260 
-251 1 
252! 



36100 

36IB1 

36864 

37249 

37636 

38026 

38416 

38809 

39204 

39601 

40000 

40401 

4t)804 

41209 

41616 

42026 

42436 

42349 

43264 

43681 

44100 

44621 

44944 

45369 

4.6796 

46226 

46666 

47089 

47624 

47961 

48100 

48841 

49284 

497-^9 

60176 

50626 

61076 

51529 

51984 

6-2441 

.52900 

63361 

53324 

61289 

54766 

56-226 

06696 

56169 

66641 

57121 

57600 

58081 

68664 

69049 

59636 

60026 

60616 

01009 

61.60-4 

62001 

tf2600 

63001 

63504 



Sq. Iloot. Cub* Root 



6859000113-7840488,5 
09678711 13 •820-2750 5 
7077888113-8561066 6 
71890,17 13-8921440.6 
7301384 13-n-283883 5 
7414876 13-9012400 5 
7620636 U'OOOOOOOl.5 
76 4.6373 14-03.jfl888;5 
776-2392 I4-07I2173'5 



7880)99 
8000000 
81-20601 
8242 108 
8366427 
848966 4 
8616126 
8741816 
8889743 
8998912 
91233-29 
9261000 
9393931 
9.6281-2S 
9663697 
9890341 
993S375 
10077696 
10218313 
10360232 
10603469 
10648000 
10793861 
10941048 
11089667 
11-2394-24 
11390626 
11.543176 
11097083 
11862362 
12008939 
12167000 
12326391 
12487168 
12619337 
1281290 4 
1-2977875 
13I442.J0 
1331-2063 
13431 27 J 
13661919 
13824000 
13997621 
1417-2-188 
14348907 
145-26789 
14706126 
14886936 
16069223 
1626-2992 
16438249 
15626000 
16813251 
10003008 



11- 
14- 
14- 
14 

14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14- 
14 
U 
14 
14 
14 
14- 
14- 
14- 
14- 
IS- 
IS- 
16- 
15- 
16- 
15- 
16- 
16- 
16- 
I5' 
15' 
15' 
16' 
IS- 
IS 
16' 
15 
15 
15 
15 
15 
15 
15 
lo 
16 
15 
15 
15 



1 067360 15 
14213666 
1774469:5 
21-26704 5 
2478068:5 
2828669 i5 
317821 1 15 
35-2700115 
387494615 
42-22051 5 
4,5683-2315 
4913767 
6258390 
6602198 
.59451it5 
6237388 
6628783 
69693t 
730<)199 
7648231 
7980436 
8323970 
8660637 
899(i644 
9331815 
■9666-296 
0000009 
0332961 
-06fi6192J6 
• 0996689 i.i 



1327460 
1657609 
1986842 
2315402 
2643375 J6 • 
•297068616- 
.3297097 6 • 
362-2916 6 - 
3:11801316- 
4-272486 16 - 
4696248 '6 - 
491933 I '6 ■ 
624 1747 !6 - 
.6663492 i-ii - 

• 588 1673 16 • 
■6-2019910- 

• 66-247 .68l6 • 
■681387l|«- 
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-74b0ii>7;6- 
•7797338|6- 
•311388316- 
•84'29795 6^ 
•8745079 6- 



748897 
758965 
768998 
778996 
788960 
7P,S890 
808786 
818618 
8-28476 
838272 
818036 
-857766 
867464 
877130 
886766 
89636S 
906941 
916 ISl 
924993 
934473 
943921 
963341 
96-2731 
972091 
931426 
0907-27 
OOOOOO 
009244 
018463 
0-27660 
■036811 
04.)913 
0.66048 
0641-26 
•073178 
-082201 
-OOni;) 
-10017(1 
-109116 
•118033 
- 126926 
- 136792 
-1 41634 
•163419 
-16-2239 
•171006 
-179747 
13.J463 
197164 
20682 1 
21446! 
22 31 18 i 
2;;i6?ii 
2 40261 
243800 
267321 
266826 
274306 
■28076() 
291194 
299604 
307993 
316359 



,Cubi Rnot 



,88 5-748807 
.'>0,5'7.V<i»(l5 
i6iJ6-7t5rf!W8 
,40.6-77H9!»0 
isn 8 •7889(50 
,00&'7PSH90 
I00l5-80H786 
18H;5'8|8(U8 
i73!5-8iSl7ti 
ltiOj5-83Hi7'i 
lu(iij'Hl8()3i) 
100!o-8fl77ii(; 
04 3-b67l<)1 
Ib8|6-87713(» 
i69i»- 88076 "i 
!ll|5-896.'ttis 
)01io-90i)941 



1-23 1 
'ti' . 
190 i 

198 I 

li»5i 
188 I 
f83i 
186 I 

199 I 

!;u 

186 1 
471) I 
)87 
144 
Slnl 
i'JJ I 
JO!) I 
JO J I 
U)i I 

j89 •;• 

400 0- 
309 6' 
J4'2 0- 
402 0- 

Uft lo- 
ss.') I ()• 
097 • 

iM;") 0' 
Jiriis- 

486 !0 • 
248 0- 
331 !8 • 
747!0- 



9l,ilti;i 

924993 
934473 
943921 
9..3341 
90-2731 
972091 
5 -98 1 420 
090727 
fi-OOOOOO 
6-009244 
6-018403 
6-0->70.'>0 
6-030811 
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0-0.i;-)04S 
0-064120 
6-073178 
6-082201 
0-00n<i;) 
0-10017(1 
1091 1.> 
ll5()3;i 
1-2092.1 
136792 
141034 
lf)341i» 
102239 
171096 
179747 
18.i403 
1971.'i4 
•20.-)82 1 
■214464 
223ii.Si 
'2;;i0?ii 
C.73l6-'2 4i)261 
991 6 -'248800 
76h'0 -267324 
87 1|« -206826 
330j6 -27430 
ii)7 6--280i'()(J 



338 

883 
796 
079 



6-291194 
6-299604 
6-307993 
6-310359 



8<iUAnE% CITDES, AND ROOTS. 



879 



No 

26a 


.Si|iiare. 


Cube. 


Sq. Ibiol. 


Cube Root 


No, 


S<]ii:tri. 


Ciil«. 


.Sq. Hoot. Ciib» Iliioi 

1 


0100!) 


10194277 


Ij-n0.)9737 


6-321704 


310 99860 


31.66 1496 1 7 -7763888 0-81 1-28 1 


ijl 


61610 


10387004: 16-9373770 


0-3230-20 


317 100489 


aiH;V.(H;j 17 -tio 11938, B-818102 


'i<y> 


66026 


I068l:i7o,16-90,-(719» 


0-3413-20 


318 101121 


32 1 67132 1 7 • 83 26646 ;0 • 82502 1 


■iM 


06630 


10777216 lO^OOOOOOO 


6-349001 


319 1(11761 


32IOI7.69,17^800.67II|0^83-2771 


•ii>i 


0()049 


10974,!93 


160312196 


6-3.>7a61 


3-20 102100 


32708000, 17 -888543810 - 83990 1 


268 


0060 1 


17173612 


16-0623784 


0-300096 


32l'l03Oll 


33076 1 1 ! 1 7 - 9 1 0472916 - H4702 1 


269 


C70S1 


17373979 


16-09347(^9 


0-374311 


3-22 1030.-) I 


33380-248; 17 9 113684 41 • 8641 21 


200 


67600 


17676000 


16-1246166 


6-382604 


323^104329 


33(;98-207 17-97-2-2008 6-801212 


201 


08121 


17779.681 


Ifi- 16649 14 10-390670 


324 104970 


340 1 2224 1 8 • 0090000 6 - 808-286 


202 


08044 


17 9847 •is 


10^1864141 


6 - 398828 


3-25' 106025 


3 1328 1 25! H • 0277564 ■ 875344 


203 


69109 


18191447 


10 -21 7-2747 


6-40(J96,-^ 


3-20 100270 


34iJ4,M»76ilH055470l 6-H8-2388 


201 


09090 


IS399744 


10 "2480768 


6-41 )0i)8 


327 ,100.) 29 


34966783, 180831 413 6-889419 


20.. 


70226 


18009626 


16 •2788-206 


0-423168 


3-28; 107684 


36287562 


18- 1107703 0-890436 


200 


707.60 


18821090 


1 6 •SOOoim 0-4312-28 


.1-20 108241 


a.iO 11289 


18- 13.83671 6 903436 


267 


7128:1 


19034103 


10 •34013 40 


0-439277 


330! 108900 


36937000 


13- 10.69021 0-010423 


208 


71821 


19248832 


10 3707066 


6-447306 


33lil09.)01 


30204691 


18-19340610-917396 


20,1 


72301 


19106109 


16^4012196 


6-466316 


332' 110221 


30694308 


13-2-208072 0-9-24356 


270 


72900 


19083000 


10^.1316707 


6-403304 


333410889 


30920037 


18--24.S287616-931301 


271 


73441 


1 990^26 11 


10-46-20776 


6-471274 


3341111566 


37269704 


18-27.66669 6-933232 


272 


73984 


201-2304.3 


16 •■4924-226 


0-4792-24 


335 11-2226 


37696375 


13 •30.3,)062 6-946149 


273 


74629 


20;l40417 


16-6227116 


0-487164 


336; 11-2890 


37933066 


13- 3303028 !6-S.5-2063 


274 


7.J076 


20670824 


16-56-29464 


0-49.)006 


337 1 113609 


38272763 


18- 367669316 -953943 


276 


76026 


20790876 


l6-583l->40 


0-50-29.60 


338,111244 


3S014172 


18-3347763 6-96.6819 


270 


70170 


2102467 


16-613-2477 0-510830 


339:111921 


38963219 


18-41 196-2610 -97-2683 


277 


70729 


2l2639;i3 


10-0133170 0-518681 


3 10| 116(300 


39304000 


18-4390889I0-979632 


278 


772S1 


21484962 


16-6733320 0-520619 


341|110281 


39661821 


18-4{;01863;6-9363f)8 


279 


77841 


21717039 


16-7032931 0-534336 


342i 110964 


40001088 


l3-4rJ32.4-20|0-993191 


280 


78400 


219;)2000 


16 ■733-2006 0-542133 


343^17649 


.10363007 


18-520269-J7-000000 


281 


78901 


•221 8804 1 


16-7030640 


O-64991-.i 


344:118330 


40707684 


13-547-2370i7 -006796 


282 


79624 


224-26708 


10-7928660 


6-657672 


3161119026 


410(J36-26 


18-5741766i7-013579 


283 


80089 


22006187 


16-62-20038 


6 -.6064 16 


346! 119716 


41121730 


18-00l07.62|7-0-2O349 


284 


80060 


22906304 


10-8.5-22996 


8-673139 


34711-20109 


41781923 


18-6279360, 7-0-27106 


286 


81226 


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10-8819430 


0-580844 


348] 121 101 


42144192 


1 3 •6647.i81 17-033860 


280 


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10-9116346 


0-588632 


349! 121811 1 


42.608649 


18-68l6417i7-040581 


287 


82369 


23639903 


10-9410743 


0-690202 


350 1122600 


4-2876000 


18-7082 6917-047-298 


288 


829-1.1 


23887872 


16-970.66-27 


0-003864 


361 : 123201 


43243661 


18-7349940 


7-0.)4004 


289 


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211 37609 


17-0000000 


0-611489 


3.6211-23904 


43614->03 


18-76 10630 


7-060696 


290 


84100 


24389000 


17-0-293804 


0-619100 


363 124609 


43980977 


18-788-2042 


7-007370 


291 


84681 


24042171 


17 •0687-221 


6-626705 


354 '12.6310 


41301804 


18-81.4.8877 


7-074044 


292 


8620.4 


24807088 


17-0880076 


0-634-287 


356|l-260-26 


44738376 


18-3114437 


7-080099 


293 


86849 


26163767 


17- 117-24-28 


6-641852 


366:120730 


46118010 


13-8079023 7-067341 


291 


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17-1464'282 


0-649399 


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18 •8914130! 7 -09.3971 


296 


870-26 


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17-1756640 


0-060930 


3.68128104 


4588-2712 


18-9203379 7-100688 


•290 


87610 


2693 1330 


17-2046506 


0-004444 


359 1128881 


40203279 


18-947-29.>3'7- 107194 


297 


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0-671940 


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■298 


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19-0000000 7-1-20307 


299 


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17-2910105 


6-686832 


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19-0202976;7- 126936 


300 


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17-3206081 


6-694329 


303I 131769 


47832147 


19-062.5.68917-133492 


301 


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27270901 


17-3493610 


6-701769 


304!l32496 


482-28644 


19-0787340,7-140037 


302 


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17-3781472 


6-709173 


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303 


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19-1333261 7-166096 


306 


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28934443 


17-5214156 0-74,6997 


370,130M0 


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19-23.63841 17 -1790.64 


308 


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17-5499288 0-763313 


371 1137641 


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19-2013603 7-185516 


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19-339079017-204332 


312 


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19-3649167|7-211248 


313 


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17-6918060 


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19-390719417-217652 


314 


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8QUAUR«, CUnKB, AND ROOTS. 



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379 IIMdil 
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419 
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159 210881 

160 211600 
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170 1220900 
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103823000 
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475 225625il07171875 
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476 
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479 
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SQUARES, cirnra, and roots. 



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•040905 
•045041 
•049114 
•033183 
•057248 
•061 310 
•005307 
•069422 
•073473 
•0775-20 
•081003 
•085003 
•089039 
•093672 
■097701 

1017-20 
10574s 
109706 



SQUARE.*, CUIJES, A.V5 ROOTS. 



383 



Nn, Square. I Cube. 



7.57 573049 
758 574.564 
7591576081 



577600 
579121 
580644 
58-2169 
583696 



433798093 
43551951'2 
437246479 
438976000 
440711081 



P(). Rool. 



•27-6136.330 

:27- 6317998 
27-5499546 
27-6680975 
27-5862284 



Cube Uotii 



442450728 27-6043475 



444194947 
- 445943744 

580225 447697125 
586756 449455096 
586-289 451217663 



589824 
591301 
592900 
594441 
595984 
597.529 

774 599076 

775 600625 



776 



002176 



603729 
605W4 
606841 
008400 
609961 
611524 
013089 



452984832 

454756609 

456533000 

45.- ■^14011 

460(»i, >648 

401839917 

463684824 

465484375 

467288576 

469097433 

470910952 

472729139 

474552000 

476379541 

478211768 

48004808 



27-6224546 
27-6405499 
■27-6586334 
27-6767050 
27-6947648 
27-7128129 



117793 
121801 
125806 
9-1-29806 
9-133803 
9-137797 
9-141788 
9-146774 
9-149757 
9-163737 
9-167714 



No.iSqn.-ire.l Cube 



9-113781 820 |67240e|65I 368000 
"•' ■■'^"- 821 1674041 155.3387661 



Sq, Rool. {Cube Root 



4855876") 28 -035691 5 



4874434i»3 



614656 48 1S90.3O4 

616-225 4837366-*i 

017796 

619309 

020944 

622521 
624100 
625681 
627264 
628849 
630436 
032025 
633616 



27-7308492 9-161606 

•i7- 7488739 9-165656 

27-7668868 

27-7848880 

27-80-28776 

27-8208555 

27-8388218 

27-8.567766 

27-8747197 

27-89-26514 ., 

27-9105716 9 



9-1696-22 

9-173585 

9-177644 

9-181500 

9-185453 

189402 

193347 

197289 

:01229 



27-.Q284301 9-205164 



27-9463772 
27-964-2629 
27-9821372 
28-0000000 
28-0178516 



797 635209 
" 3 036804 
) 038401 
) 040000 
1 6U601 
! 043204 
I 644809 
046416 
I 648025 
' 649636 
051249 
662864 
054481 
656100 
657721 
659344 
813 600909 
'^14 6«2596 
31 -1664225 
816,6058.56 
817 



489303872 
491109069 
493039000 
494913671 
49679'>'^88 
498077 -io7 
500566184 



23-0535^203 



•23-071.3.377 
28-0,891438 
28 -J 069380 
28- 1247^222 
•28-14-24940 
23-1602557 
23-1780056 



502459375 28-1957444 
504358330 28-2134720 
506-26157.3 28-2311884 
508169592 28-2-188938 
51008-2399 23-2665881 
512(JD0000 28-284-2,712 
513922401 28-3019.434 
515849008 28-3196045 
517781027 28-3372546 



818 
819 



019718464 28-3548938 
021660125 28-3725219 
023006616-28-3901391 
520557943 28-4077454 
527514112 28-4-253408 
529475129 28-4429253 
531441000 '28-4004989 
53341173128-4780617 
5353873-28 28-4956137 
537367797 28-5131549 
539353144 28-6306852 
541343376 28-6482048 

o4333«4»6 23-6657137 i,-<ytvioi 

667489 54533851 3128 • 58321 1 919 - 343473 

----_-. ...,.,,.. ._..,..,^u3:tj;? ja;;»BD 

670.oi|549353369J28-6181760 9-8»«O95 



9-209096 

9-213026 

9-215950 

9-2^20873 

224791 

228707 

23^2619 

23752S 

240433 

244330 

2482.34 

252130 

256022 

259911 

!)• 263797 

9-267680 

9-271559 

9-275435 

9-279303 

9-28317S 

9-287044 

9-290.907 

•294767 

•298624 

302477 

3063-28 

810175 

314019 

9-317860 

9-321697 

9-325532 

9-329363 

9-333192 

9-337017 

9-340838 

9-344667 



822 675684 
823677329 
824678976 
825J680625 
820i682276 
8271683929 
828 1 685634 

829 i 687241 
8.30 688900 
831 1690061 
832|6922i24 
833!693889 
334690056 

830 697220 
830693390 
8371700569 



28 



5504122-18 
657441767 
659476224 
561515626 
563559976 
665609283 
067663553 
569722789 
.571787000 
673806191 
576930308 
578009637 
680093704 
582182876 
584277056 
- 586376203 

838 702244 588480472 

839 703921 590589719 

840 706600 592704000 

841 707281 594823321 

842 708964 590947688 
813 710649,599077107 

844 712330 601211584 29 

845 714025J603351126 29 
846|715716j605495736 29 
8'*7 717409 607645423 29 
848 719104009800192 29' 
3 19|720801 01 1960049 -29' 
8o0 72-2500 014125000 -29- 

"" 29- 



-24201 
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28 
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28- 

28- 

28 

28 

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016-295061 1.^., 

6184702081-29 



727609 620000477 
729316 022835864 
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73-2736|027222016 
734449|629422793 
736164 631628712 
737831 033839779 
739600,036056000 
741321038277381 
743044 640003928 



744769 

746496 

865748-225 



042735647 
044972644 

647214626 

866 749900 049461896 



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807 751689 651714363 

868 753424 663972032 

869 755161 656234909 
370|756e0O 65350.3000 

758641 66077631 1 
760384 663054848 
762129 665338617 
63376 667627624 
65626 669921876 
707370|672221376 
709129 674520133 
770884 6768.36162 
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880)774400j631472000J29 
Cioii770ioli0837y?»fjj J29 
88-2^777924|686128968 29 



871 
872 
873 
874 
875 
876 
877 
878 



9 
29 
29 
29 
29 
29 
29 
'29' 
■29- 
29- 
29 
29 
29 
29 
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29' 
29- 
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29 
29 
29 
29 
29 
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29 



■6356121 !9 
6530970 9 
6705424 
•6879766 
•7064002 
-7228132 
-7402157 
•7670077 
•7749891 J9 
■7923601 9 
■8097206 
■8270706 
-8444102 
•8617394 
-8790682 
•8963606 
■9236640 
9309023 
9482297 
-9654907 
-9827635 
-0000000 
■0172363 
■0344623 _ 
■0516781 9 
•0688837 9- 
0860791 9 
103-2644 9 
12043969 
•1376046 9 
•1647595 9 
•1719043 
•1890.390 
•2061637 
■2232784 
■2.103830 
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274.5623 
2916370 
•3037018 
•3257666 
•34'28015 
•3698.365 
■3768616 
■3933769 
■4108323 
4278779 
4148637 
4018.397 
4788059 



•4957624 
■5127091 
6296461 
6466734 
-5634910 
•5803989 
•6972978 
•6141858 
■6310648 „ 
■647932519 
66479391a 
681644219 
69848489 



359902 
303705 
367005 
371302 
37.5096 
378837 
382676 
380460 
390212 
394020 
397790 
401509 
405339 
409106 
412809 
410030 
420387 
424112 
427894 
431642 
435333 
439131 
442870 
440007 
450,341 
461072 
457800 
401520 
465247 
463906 
472632 
476396 
480106 
483813 
4876181 
491220 
494919 
498616 
602308 
•605998 
■609686 
■513370 
517051 
520730 
524406 
528079 
531749 
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■539082 
5427.44 
546403 
550059 
663712 
•507363 
■561011 
•664656 
608298 
671938 
675574 
679208. 
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686468 { 
6900941 

1 



9- 

9^ 

9- 

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9- 

9- 

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0- 

9 

9 

9 

9 

9 

9 

9- 

9- 

9 

9 

9 






SQT>fcRES, CUBES, AND ROOTS. 



\o. 



Scjuare. 



Cuba 



't3 7796SS1 G88.l6o387 
>84 781466 0f)0807I04 
85 78332.5 15<)3151125 
30 784990 6ft56064.56 
o87 786769 697804103 
;-i68 788344 700327072 
189 790321 702595369 
890 792100 704069000 
191 793881 707347971 

892 796604 709732288 

893 797449 712121957 

394 799236 714516984 

395 801025 716917375 

896 802816 719323136 

897 804609 721734273 

898 806404 724150792 

899 808201 726572699 

900 810000 729000000 

901 811801 731431701 

902 813604 733870808 

903 815409 736314327 

904 817216 738763264 

905 819025 741217625 

906 820836 74.3677416 

907 822649 746142643 

908 824464 748613312 

909 826281 751089429 

910 828100 753571000 

91 1 829921 756058031 

912 831744 768550528 

913 833569 76104849 

914 83-5396 763561944 
916 837225 766060875 

916 839050 788375296 

917 840389 771096213 

918 842724 773620633 

919 844661 776151559 

920 846400 778663000 

921 3^8241 781229961 

922 S50084 783777443 
923i851929 786330407 



Si}. Hoot. 



29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29' 
29 
29' 
29- 
29' 
29- 
29- 
30- 
30- 
SO- 
SO- 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30' 
30' 
30' 
30' 
SO- 
SO- 
30- 
30- 
30 



■7163159 
■7321375 

•7489496 
■7657521 
•7825452 9 
■79932S! 
8161030 9 



Cuba Root 



8328678 
•8496231 
■8663690 
■8831066 
•8998328 
'9105506 
9332591 
9499583 
9666481 
983.3287 
•0000000 
■W 66620 
•0333148 
•0499584 
•0666928 
•0832179 
•0998339 
•1164407 
■1330383 
•1490269 
•166206319 
•182776519 
■1993377 9 
■2158899 9 
■2.324329|9 
■248966919 
■26.5491919 
2820079] 9 
■29851489 
31501239 
•33150189 
' 347981 :j 9 
364452&,9 
380915119 
3973683 9 



92 1|853776 788889024 
935 8556-25]791453126 30^4138127 9 
9-J6 857476;79402-2776 30^430'3481 9 
937 8593-39 798597933 30-4466747 9 
933|801184 799178752 30-4630924 9 
939:863041 801765089 .30-4796013 9 
930|804900 804357000 30^4959014 9 
93l|86676180695449l!30^61229-36'9 
932i868634;S09557568;30-5286750|9 
933jS704b9.3121663.'i7j30-5450487 9 
931|872350|814780504'30-C614136 9 
935 874-2-35;8174O0375i30-6777697 9 
936:876096, 830036856|30 • 6941 1719 
937|877S69,S-33656953 30^6104657 9 
938;879844| 635293672130 • 6-307857 9 
939.t;8172l|837936019j30-6431069'9 
'140 883600:830584000 30-6594194'9 
41 885481.83.3237621 30-6757233'9 



-593716 
•597337 
•600956 
•604570 
•608182 
•611791 
•015398 
■619003 
•623603 
•626201 
•629797 
-633390 
■636981 
■640569 
■644164 
647737 
■651317 
'654894 
6.58468 
663040 
665609 
669176 
672740 
676302 
679860 
683416 
086970 
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•697615 
•701158 
•704699 
■708237 
•711772 
•715305 
■718S35 
■722363 
•725888 
•729411 
■732931 
■736448 
■739963 
•743476 
•746986 
•750493 
•753998 
•767500 
'761000 
'764497 
•767993 
771434 
'774974 
778462 
'782946 
785429 
788909 
79238e 
796861 
799334 



Wo. 



S<|uaie 



942 
943 
944 
946 
946 
947 
948 
949 
950 
961 
962 
963 
954 
985 
956 
967 
958 
959 
960 
961 
962 
963 
964 
965 
966 
967 
968 
969 
970 
971 
972 
973 
974 
975 
976 
977 
978 
079 
930 
981 
982 
983 
984 
986 
986 
987 
988 
989 
990 
991 
992 
993 
994 
996 
996 
997 
998 
999 
1000 



887364 
889249 
891130 
893025 
894916 
896809 
898704 
900601 
902600 
904401 
906304 
908209 
910116 
912025 
913936 
915849 
917764 
919681 
921600 
923521 
925444 
927369 
929296 
931225 
933156 
935089 
937024 
933961 
940900 
943841 
944784 
946729 
948676 
950628 
952576 
954529 
966484 
958441 
960400 
962301 
964324 
966289 
968256 
970225 
972196 
974169 
976144 
970121 
980100 
982081 
984064 
986049 
988030 
990025 
99-3016 
994009 
996604 
998001 
1000000 



Cubs. 



Sij. Root. 



Ciihe Koot 



636896888 

838501807 

841232384 

84.3908625 

846590536 

849278123 

861971392 

854670349 

857375000 

860085361 

862801408 

86.5523177 

868250664 

870983875 

87372-2816 

876467493 

879217912 

881974079 

884736000 

887603681 

890277128 

693056347 

895841344 

89363212.= 

901428696 

904231063 

907039232 

909863i*i<9 

912673(».0 

915493(),1 

918330C48 

921167317 

924010424 

926869376 

9-39714176 

932574833 

935441362 

938313739 

94119-2000 

944076141 

940966168 

949862087 

952763904 

956671626 

958636256 

961504803 

964430272 

967361669 

970299000 

973-342271 

970191488 

979146657 

982107784 

985074875 

938047936131 

991026973131 

994011992131 

997002999 31 

100000000031 



30 
30 
30 
30 
30 
30 
SO 
SO 
30 
SO- 
SO- 
SO- 
SO- 
30- 
30- 
•St 
30 
30 
30 
31 
31 
31 
31 
31 
31 
31 
31' 
31' 
31' 
31' 

31 ■ 

31- 

31- 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31' 

31' 



•6920185 
•7083051 
'7245830 
■7408523 
7571130 
•7733651 
•7896086 
•805ai36 
■ 8-320700 
•8382879 
•8544972 
•8706981 
■8368904 
•9030743 
■'■ 193497 
9354166 
•9515761 
■9677251 
•9838663 
0000000 
0161248 
0322413 
0483494 
0644491 
0805405 
0966236 
1126984 
•1287648 
•1448230 
■1608729 
•1769145 
•19^39479 
•2089731 
•2249900 
•2409987 
•2669992 
•27-39916 
•2889787 
•3049517 
•3209195 
•3368792 
•3528308 
■3687743 
•3847097 
•4006.309 
•4165561 
•4324673 
•4483704 
■4643664 
■4801525 
■4960315 
•6119025 
•6377665 
■5436206 
5594677 
'.>r.5.3068 
6911380 
6069613 
6227766 



9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9' 
9- 
9- 
9^ 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9- 
9- 
9- 
9- 
9- 
9- 
9^ 
9^ 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9' 
9' 
9- 
9- 
9- 
9- 
10- 



•8O3804 
'806'r/l 
80973O 
813199 
•816655 
•820117 
•8-33572 
•8-37035 
•830476 
•833934 
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■844234 
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■851128 
■854562 
857993 
•8614-32 
•864848 
•868373 
•871694 
•876113 
•878530 
•881945 
•885367 
•888767 
•892175 
■895580 
■898983 
■902333 
•906782 
•909178 
912671 
916962 
919351 
922738 
926122 
•9-39504 
•932884 
•936261 
•939636 
•943009 
•946380 
•949748 
•9.531141 
•956477 
•959839 
•963198 
-9665,55 
•969909 
■973363 
■976U12 
•979960 
983305 
9866 H) 
989990 
993339 
9%'666 
000000 



Nu. (iij 
\iy. 



1 

2 
3 
4 
5 

6 

7 

8 

9 

10 

11 

13 

13 

14 

1,5 

l(i 

17 

19 
30 
21 

33 
T.i 

•Ji 



I 



Sij. Root. 



Ciihe Koot 



M- 6920 185 

30-7083051 

30-7245830 

30-7408523 

30-7571130 

30-7733651 

30-7896086 

30-8058-136 

30-8320700 

30-8382879 

30-8544972 

30-8706981 

30-8868904 

30-9030743 

30 •'•192497 

3* 9354166 

30-9515751 

30-9677251 

30-9838668 

31-0000000 

31-0161248 

31-0322413 

31-0483494 

31-0644491 

31-0805405 

31-0966236 

31-1126984 

31-1287648 

31-1448230 

31-1608729 

31-1769145 

31 -19-^9479 

31-2089731 

31-2249900 

31-2409987 

31-2569992 

31-27-29915 

U -2889767 

11 -3049517 

(1-3209195 

11-3368792 

11-3528308 

11-3687743 

11-3847097 

(1-4006309 

(1-4165561 

H -4324673 

il -4483704 

H -4642664 

il -4801525 

1 -4960315 

1-6119025 

1-6277665 

1-5436206 

1-5694677 

1 -.1753068 

1-6911380 

1-6069613 

1-6227766 



9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9- 
9- 
9- 
9- 
9- 
9- 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9' 
9 
9' 
9- 
9- 
9- 
9- 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9' 
9' 
9- 
9- 
9- 
9- 
10- 



•8O'.'M04 
- 8062V 1 
-809736 
813199 
-8I66.-)5 
-82011; 
•823072 
•827025 
•830476 
■833924 
•837369 
•840813 
•844254 
847692 
851128 
854562 
-857993 
-86 1422 
-864S48 
•868272 
-871094 
-875U;} 
•878630 
•881946 
•885367 
•888767 
•892175 
-895680 
•898983 
■902383 
■906782 
■909178 
912671 
916962 
919351 
922738 
926122 
•929504 
•932884 
•936261 
•939636 
•943009 
•946380 
•949748 
•963114 
•966477 
•959839 
-963198 
-966555 
■969909 
•973262 
■976012 
■979900 
983305 
986649 
989990 
993329 
996666 
000000 i 



TABLES. 



3S'> 



•^'^^ ^^- ^^ T"^^- AMOU NTS OF £1 AT COMPOUND INTEREST. 

3 per cent 



Mo. of 
I'ay- 
mcnt] 



4 per cent 



1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
26 



per cent 



1-03000 
1-00090 
1-09273 
!• 1-255 1 
1-169'27 
1-19405 
1 -22987 
1-26677 
1 -30477 
P343!42 
1 •38423 
1 •4-2576 
1 •4U853 
1 •51259 
r 65797 
1^60471 
1 •65285 



70243 
75351 
806 11 
860-29 
91 610 
1-97359 
2-03279 
2-09378 



l: 

1- 

1^ 

1- 

1- 

1- 

!• 

1- 

1- 

1- 

1- 

1- 

1- 

1- 

1- 

2- 

2- 

2- 

o. 



04000 
08100 
1-2486 
16986 
21063 
26532 
81593 
36S5T 
4-2.331 
48024 
53945 
■60103 
06507 
73168 
80094 
87298 
94790 
02582 
10685 
19112 
27877 
2-3o!Jri2 
2-40472 
2-66330 
2-60584 



1 •05000 

!• 10250 

1^ 16762 

1 •21551 

1^276^28 

!• 34010 

1^40710 

1-47745 

1^56133 

1 •62889; 

I • 71 034 

1-79586 

1-88565 

1-97993 

2-07893 

2-18287 

2-29202 

2-40662 

2 •5-2695 

2-65330 

2-78596 

2-925-20 

2-07152 

3-2-2510 

3-38635 



6 per cent 



•06000 
•12360 
•19102 
-26248 
■338-23 
•41852 
■50363 
59385 
68948 



Vo. of 

Pay 

meiita 



1-79086 
-89830 
•012-20 
•13-293 
•26090 
•39656 
■54035 
•69277 
85431 
02560 
20713 
39956 
3-60354 
3-81975 
4-01893 
4-29187 



20 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

43 

49 

60 



3 per cent 



2-16659 
2-22129 

2-28793 

2-35657 

2-42726 

2-50008 

2 '57508 

2-05233 

2-73190 

2-81386 

2-89828 

2-98523 

3-07478 

3-16703 

3-26204 

3-35990 

3-46070 

•56452 

67145 

78160 

89504 

01 190 

13-225 

25622 



■i per cent 



4-38391 



77-247 

88337 

99870 

11865 

24340 

37313 

3 •50806 

3-64833 

3-79432 

3-94609 

10393 

26809 

43881 

616371 

80102 

99306 

19278 

40049 

5-61651 

6-84118 

6-07482 

6-31782 

6-57053 

6-83335 

7-10663 



5 per c«nt 



6 
5 
6 
6 
6 
7 
!■ 
7- 
8- 
8- 
8- 
9 
9 
10 
10 
11 



-56567 
-73346 
-9'2(ll" 
-11614 
-32191 
-53804 
■76494 
•00319 
25335 
51601 
79182 
•08141 
•38548 
•70476 
•03999 
•39199 
•76159 
•14967 
•5671.' 
•98,501 
434-26 
90597 
40127 
92183 
46740 



6 per cpiil 



6 
6 
7 
7 
8 
8 
9 
9 

10- 
10- 
11^ 
12- 
12- 
13 
14 
16 
16 
17 
18 



■64933 
8-2235 
] 1 11)9 
41839 
74319 
•08.^10 
•453;t9 
•810:>LI 
■25 1(1 2 
'6StUtJ 
I 1725 
fi3(l(rj 
15125 
70351 
•28572 
■90280 
•55703 
•'23045 
■9.3543 
■76401 
■59049 
•46592 
39387 
37750 
42015 



J'ABLE OK THE AMOUNTS OF AN ANNUITY OF 



£1. 



Su. (ilj 1 

P.iy. Spercenl. <pcrci;iit| operctfiit 



1 

2 
3 
4 

5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
l(i 
17 
IS 
19 
20 
21 
22 
•23 
•2i 



I 



6 
7 
8 
10 
11 
12 
14- 
15' 
17- 
18- 
20- 
21- 
•23- 
25 

:i) 

28 
)0 
!2' 
il- 



-oonoo 

•03000 
■09l)i)0 
-J 8363 
-30913 
•40841 
• 66246 
•89231 
■15911 
■463t*S 
80779 
19203 
61779 
08632 
59H!)! 
!M-!8!^1 
7(il.i'.tl-23 
'iU13l25 
lH>'i7J27 
•"'7(!37 -29 

OTorsUi 

53{i7Sj34 
4.V2s,-ii3(i 
4-6l7J3:l 
4^y20'41 



1 

2 
3 

4 

6 

6 

7' 

9' 

10- 

li- 

13- 

15- 

16- 

13 

20 



-00000 
■((4000 
-12I60i 
•■24610 
-41032 
-63297 
■89829 
•21423 
58279 
00611 
4vS63; 



1 



6 per eentl' 



00000 
05000 
15250 3 
31012 4 
62563 5 
80191 6 
14201 8 
9-54911 9 
11-0-2656 11- 
12-57739 13- 
M- 20679 14- 
02580 15-91713 16- 
6-26Sl|l7^7i29!i 18- 
29191 |19*?)9S63 21 • 
02359-21 •o7S.-i6 23 • 



3 per cent 



■« per cent 6 per cent 



•S-2J.03''23-65749 
■69751 25-84037 
■61541.-28-I323S 
671-23 00-53900 
778(»,M 33 •06595 
96920 35 •710-.).:, 

■2171)7 ,{8-50521 
0I7.'-D41-.130!/ 
(lo'iiid ■I4^5(12U0 



25 -( 
28--. 
30-f 
33-7 
36-7 
39 - P 
43 -S 
fti-;i 

.50 ■ S 



6-loWl 4V-,;.'71tli54 



50 



18-66304 

SO -70963 

2-93092 

5-21885 

7-57641 

0-00268 

2-50276 

5-07784 

7^73018 

»• 46208 

3-27594 

3-17422 

M5945 

2-23423 

>-40126 

?• 66330 

!-02320[l04 

'•48389 110 

I-048U 115 

!-719y6 121 

■•50146'l26^ 

• 396.50 '132 - 

• !0.S39,139- 
•5106,V|I5- 
■796tt7jl5'2 



44 

47 

49 

52 

66 

69 

62 

66- 

69- 

73 ■ 

77 • 

81^ 

85- 

90 

95 

99 



•31174 

•08421 
•96758 
•966'29 
•08494 
•32833 
■7QU7 
■20953 
85791 
65-222 
59831 
70225 



61 

64 
68 
62 
66 
70 
75 
80 
85' 
90' 
95- 
101 



•11345 
•60913 

•4025S 
•32271 
•43886 
•76079 
•29329 
•06377 
■0669t; 
32031 
8363i' 
628 M 



6 percent 



o9 
63 
68 
73 
79 
81 
90 
97' 

104 

111 

119 

J 27 



97034| 107 •70954 135 
•09502 145 • 
•79977 154 • 
•8397(;i65^ 
'•23175 175- 
■99334 187 



114 
120 
127 



•4091 
■02551 

■82654 

■31960|l35 

01238 142 

412881151 

02939J159 

87057 168 

94539 178 

■2632i|l88' 

53.';73:li)S 

667081 209- 
I 



14300 
•7001(.. 
■68511 
■11942 
025.3'> 
42(iGu 
a47Pii 



199 
212 
22(i 
211 

256 

■?2 
00 



•15638 
•70576 
•62311 
■03980 
■05819 
■S0168 
88978 
•34316 
•18375 
•43478 
•12087 
•26812 
■904-20 
•05846 
•76196 
04768 
95054 
•6075e 
■76303 
•74351 j 
■50812 
09861 
56453 
95840 
33590 



Ml 
III 



386 



TABLES. 
TADLR OF THE PRKSKNT VAM/F'.S OV AN AN'XITITV 



-Vo. of 
IMy- 



I 

3 

a 

4 
6 
6 
7 
8 
9 
10 
11 
li 
13 
14 
IS 
16 
17 
18 
19 
20 
21 
22 
23 
34 
33 



OF £1. 



3l'erc«iit j<i,trcBm 



f' larceiitl i)»r cant 



0-970&7 
l-!)i;)-ir 
2-8:28(il 
3-7J71() 
4-Wt)Tl 
a-41719 

7 -01 oaf) 

7- 78611 
8- 53030 



9-95400 
10-63490 
U -29607 
U- 93794 
12-66110 
13- 1661-2 
13-76351 
14-323S0 
14-8774,-! 
15 •41602 
15-93692 
16-44361 
16-93664 
17-41316 



o-aois-i 

l-8t-i6l9 

2-77519 

3-6^999 

4-4dl)^i 

5-a4214 

6-00-.>0o 

fi- 73274 

7-43533 

8-1J089 

8-76058 

9-38507 

9-98565 

10- 5631-2 

ll-11849fl0 

11-65239 10 

12-10567 

12-65940 

13-13394 

13-59032 

14-02916 

14-45111 

14-85684 

15-24696 

15 •6-2208 



-95238 

-85941 

-75326 

-54595 

-32948 

-07569 

•78G37 

•46321 

• 10782 

•72173 

30641 

86325 

3SI367 

89864 



0-94310 
1-83339 
2-67301 
3-46510 
4 -21 236 
4-91732 
6-58238 
6-30979 
6-80169 
7-36009 
7-886a7 
8-38394 
8-&-V268 
9-29498 



N'o. nt 
l>ny- 



■i per cent 



•< per car.'. 



37965 9-71-225 
83777110-10589 



27406 
-68968 
•08533 
•46-221 
•82115 
10300 
48357 
79864 
09394 



10-47726 
10-82760 
11-15811 
11-46993 
11-70407 
12-04158 
13-30338 
12-65036 
12-78335 



26 

37 

28 

39 

30 

31 

32 

33 

31 

35 

36 

37 

38 

39 

40 

41 

43 

43 

44 

45 

46 

47 

48 

49 

60 



17 

IS 

18 

19 

19 

30 

•20 

30' 

21- 

31- 

31- 

33- 

33- 

33 

33 

33 

33 

33 

34' 

34' 

'24- 

35- 

35- 

35- 

25 



•87684 
32703 
76411 
-18346 
•60044 
-00043 
-38S77 
•76579 
-13184 
•48722 
■83-225 
•16724 
49246 
80822 
11477 
41-240 
70136 



5 per cent I 6percfu» 



lt>- 98277 

16-32968 

16-^6306 

16-98371 

17-29303 

17-08849 

17-87356 

18-14704 

18-41119 

13 •66461 

IS •908-28 

19 •I 4258 

19-36786 

19-68448 

19-79277 

19-99305 

20-18563 



93190-20-37079 
30-64884 
30-7-2004 
20-88465 
21-04393 
21-19513 
31-34147 
31-48318 



-354-28 
-51871 
•77545 
■ 02471 
•36671 
60166 
73977 



14-37518 

14-64303 

14-89813 

15-14107 

16-37-245 

15-59381 

15-80367 

16-00255 

16-19290 14 

163741S 14 

16-64685 

16-71138 

16-86739 

17-01704 

17-15908 

17-29436 

17-42320 

17-64591 

17-66377 

17-77407 

17-88006 

17-93101 

18-07715 

18-16873 

18-35593 



14 
14 
14 
14 
15 
15 
16 
15 
16 
15 
16 
15 
16 
16 
15' 



I 



•00316 

■31083 

40616 

69073 

76483 

93908 

08404 

33033 

•36814 

•49824 

-62099 

•73678 

•84603 

■94907 

■04630 

13801 

32454 

30617 

33318 

46583 

62437 

68903 

66002 

70767 

76186 



IRISH CONVERTED INTO STATUTE ACRES. 



Iriah. 



SRltUU. 



R. P. 






u 






10 

30 

1 

2 

3 



A. n 















1 

3 



p. 
1 
3 

4 

6 

8 

16 

32 



r. 

n 

26 
Ml 

3 

6 
13 



Iriih. 






34 24 
9 172 



1 34 ll| 






A. 

1 

2 

3 

4 

6 

7 

S 

9 

10 



Statute. 



Iriih. 



A. 


K. 


p. 


T. 


A. 


1 


3 


19 


6 


20 


3 





38 


10 


30 


4 


3 


17 


153 


40 


6 


1 


36 


21 


50 


8 





15 


26.1 
ij 
6j 

111 


100 


9 


3 


35 


200 


11 


1 


14 


300 


12 


3 


33 


400 


14 


2 


12 


17 


600 


16 





31 


2-2i 


1000 



Statute. 



A. B. F, T 

32 1 23 14} 

48 2 16 6i 

64 3 6 28/ 

80 3 38 20l 

161 3 37 101 

323 3 34 2l| 

485 3 32 2 

647 3 29 13| 

809 3 26 23 

1619 3 13 163 



VALUE OF FOREIGN MONEY IN BRITISH, 
Silver being 5*. per ounce 



1 Florin is worth 
16 Schilliiig.s (Hamburg) 
1 Mark (Frankfort) . 
1 Franc 

1 Milree (Lisbon) 
8 Reali . . 



». d. 
1 8 

15| 

9i 
4 8 
3 IJ 



1 Dollar (Now York) . 
96 Skillings (Copenhagen) 
1 Lira (Venice) 
1 Lira (Genoa) 
1 Lira (Leghorn) . 
1 Ruble . . 



*. d. 

4 3 

2 2| 

8} 





3 






TV OF £1. 



5 per cent 6perciu(l 



14-37618 

14-64303 

14-89812 

15-14107 

16 •37-245 

15- 69-281 

15-80267 

18- 00255 

16- 19290 

16-3741S 

16-61685 

16-71128 

16-86789 

17-01704 

17-15908 

17-29436 

17-42320 

17-64591 

17-66277 

17-77407 

17-88006 

17-98101 

18-07715 

18-16872 

18-25592 



13-00316 
13-21083 
13-40616 
13-69072 
13-76183 
13-9-2908 
14-08404 
14-23023 
'14-36814 
14-498-24 
14-6209«» 
14-73678 
14-84602 
14-94907 
16-046,10 
16-13801 
16-22454 
15-30617 
16-38318 
16-45583 
16-62437 
15-68903 
16-66002 
15-70767 
16-76186 



Statute. 



A. 


B 


32 


1 


48 


2 


64 


3 


80 


3 


161 


3 


323 


3 


485 


3 


647 


3 


809 


3 


619 


3 



P. T 

23 14} 
16 6.1 

6 28j 

38 20| 

37 10| 

34 21| 
32 2 

29 123 

26 23 

13 16J 





». 


d. 




, 


. 4 


3 1 


jcn) 


o 


2 




. 


. U 


8 




. 


, 


9 




• 


. 


7 




« 


. 3 


li